asme 19.1 test uncertainty_05.pdf

A N A M E R I C A N N A T I O N A L S T A N D A R D

Test Uncertainty

ASME PTC 19.1-2005(Revision of ASME PTC 19.1-1998)

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Date of Issuance: October 13, 2006

The 2005 edition of ASME PTC 19.1 will be revised when the Society approves the issuance ofthe next edition. There will be no Addenda issued to ASME PTC 19.1-2005.

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Copyright © 2006 byTHE AMERICAN SOCIETY OF MECHANICAL ENGINEERS

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CONTENTS

Notice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiForeword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiCommittee Roster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Section 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-2 Harmonization With International Standards . . . . . . . . . . . . . . . . . . . 11-3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Section 2 Object and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-1 Object. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-2 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Section 3 Nomenclature and Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33-1 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33-2 Glossary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Section 4 Fundamental Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54-1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54-2 Measurement Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54-3 Measurement Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54-4 Pretest and Posttest Uncertainty Analyses. . . . . . . . . . . . . . . . . . . . . . 11

Section 5 Defining the Measurement Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135-1 Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135-2 Selection of the Appropriate “True Value” . . . . . . . . . . . . . . . . . . . . . 135-3 Identification of Error Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135-4 Categorization of Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155-5 Comparative Versus Absolute Testing. . . . . . . . . . . . . . . . . . . . . . . . . 16

Section 6 Uncertainty of a Measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176-1 Random Standard Uncertainty of the Mean . . . . . . . . . . . . . . . . . . . . 176-2 Systematic Standard Uncertainty of a Measurement . . . . . . . . . . . . . 186-3 Classification of Uncertainty Sources. . . . . . . . . . . . . . . . . . . . . . . . . . 196-4 Combined Standard and Expanded Uncertainty of a

Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Section 7 Uncertainty of a Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227-1 Propagation of Measurement Uncertainties Into a Result. . . . . . . . . . 227-2 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237-3 Random Standard Uncertainty of a Result . . . . . . . . . . . . . . . . . . . . . 237-4 Systematic Standard Uncertainty of a Result. . . . . . . . . . . . . . . . . . . . 247-5 Combined Standard Uncertainty and Expanded Uncertainty of a

Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247-6 Examples of Uncertainty Propagation . . . . . . . . . . . . . . . . . . . . . . . . . 24

Section 8 Additional Uncertainty Considerations. . . . . . . . . . . . . . . . . . . . . . . . . . 288-1 Correlated Systematic Standard Uncertainties. . . . . . . . . . . . . . . . . . . 288-2 Nonsymmetric Systematic Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . 318-3 Fossilization of Calibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358-4 Spatial Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

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8-5 Analysis of Redundant Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368-6 Regression Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Section 9 Step-by-Step Calculation Procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 419-1 General Considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419-2 Calculation Procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Section 10 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4310-1 Flow Measurement Using Pitot Tubes . . . . . . . . . . . . . . . . . . . . . . . . . 4310-2 Flow Rate Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710-3 Flow Rate Uncertainty Including Nonsymmetrical Systematic

Standard Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5010-4 Compressor Performance Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . 5110-5 Periodic Comparative Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Section 11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Section 12 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Figures4-2-1 Illustration of Measurement Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . 64-2-2 Measurement Error Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74-3.1 Distribution of Measured Values (Normal Distribution) . . . . . . . . . . 84-3.3 Uncertainty Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115-3.1 Generic Measurement Calibration Hierarchy . . . . . . . . . . . . . . . . . . . 145-4.3 Difference Between “Within” and “Between” Sources of Data

Scatter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167-6.2 Pareto Chart of Systematic and Random Uncertainty Component

Contributions to Combined Standard Uncertainty . . . . . . . . . . . . . 278-2.1 Schematic Relation Between Parameters Characterizing

Nonsymmetric Uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328-2.2 Relation Between Parameters Characterizing Nonsymmetric

Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348-5.1 Three Posttest Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710-1.1 Traverse Points (Example 10-1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4410-2.1 Schematic of a 6 in. � 4 in. Venturi. . . . . . . . . . . . . . . . . . . . . . . . . . . 4810-4.1 Typical Pressure and Temperature Locations for Compressor

Efficiency Determination. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5710-4.7 The h-s Diagram of the Actual and Isentropic Processes of an

Adiabatic Compressor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6110-5.1-1 Installed Arrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6310-5.1-2 Pump Design Curve With Factory and Field Test Data Shown . . . . . 6410-5.1-3 Comparison of Test Results With Independent Control Conditions . 6410-5.2 Comparison of Test Results Using the Initial Field Test as the

Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Tables6-4-1 Circulating Water Bath Temperature Measurements (Example

6-4.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206-4-2 Systematic Uncertainty of Average Circulating Water Bath

Temperature Measurements (Example 6-4.1) . . . . . . . . . . . . . . . . . . 217-6.1-1 Table of Data (Example 7-6.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257-6.1-2 Summary of Data (Example 7-6.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267-6.2-1 Table of Data (Example 7-6.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267-6.2-2 Summary of Data (Example 7-6.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278-1 Burst Pressures (Example 8-1-1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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8-6.4.5 Systematic Standard Uncertainty Components for Y Determinedfrom Regression Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

9-2-1 Table of Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429-2-2 Summary of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4210-1.2 Average Values (Example 10-1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4410-1.3-1 Standard Deviations (Example 10-1) . . . . . . . . . . . . . . . . . . . . . . . . . . 4510-1.3-2 Summary of Average Velocity Calculation (Example 10-1) . . . . . . . . 4510-1.6 Standard Deviation of Average Velocity (Example 10-1) . . . . . . . . . . 4610-1.9 Uncertainty of Result (Example 10-1) . . . . . . . . . . . . . . . . . . . . . . . . . 4810-2.1-1 Uncalibrated Case (Example 10-2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810-2.1-2 Absolute Sensitivity Coefficients in Example 10-2 (Calculated

Numerically) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5010-2.1-3 Absolute Sensitivity Coefficients in Example 10-2 (Calculated

Analytically) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110-2.1-4 Absolute Contributions of Uncertainties of Independent Parameters

(Example 10-2: Uncalibrated Case) . . . . . . . . . . . . . . . . . . . . . . . . . . 5210-2.1-5 Summary: Uncertainties in Absolute Terms (Example 10-2:

Uncalibrated Case) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5210-2.1.1-1 Relative Uncertainty of Measurement (Example 10-2: Uncalibrated

Case) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5210-2.1.1-2 Relative Contributions of Uncertainties of Independent Parameters

(Example 10-2: Uncalibrated Case) . . . . . . . . . . . . . . . . . . . . . . . . . . 5310-2.1.1-3 Summary: Uncertainties in Relative Terms for the Uncalibrated

Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5310-2.1.1-4 Relative Uncertainties of Independent Parameters (Example 10-2:

Calibrated Case) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5310-2.1.1-5 Relative Contributions of Uncertainties of Independent Parameters

(Example 10-2: Calibrated Case) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5410-2.1.1-6 Summary: Uncertainties in Relative Terms for the Calibrated

Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5410-2.1.1-7 Summary: Comparison Between Calibrated and Uncalibrated

Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5410-3-1 Absolute Contributions of Uncertainties of Independent Parameters

(Example 10-3: Uncalibrated, Nonsymmetrical SystematicUncertainty Case) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

10-3-2 Summary: Uncertainties in Absolute Terms (Example 10-3:Uncalibrated, Nonsymmetrical Systematic Uncertainty Case) . . . . 56

10-4.1-1 Elemental Random Standard Uncertainties Associated With ErrorSources Identified in Para. 10-4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

10-4.1-2 Independent Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5710-4.1-3 Calculated Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5710-4.3.2-1 Inlet and Exit Pressure Elemental Systematic Standard

Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5810-4.3.2-2 Inlet and Exit Temperature Elemental Systematic Standard

Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5910-4.7 Evaluation of Analysis Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6210-5.1-1 Pump Design Data (Tc p 20°C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6310-5.1-2 Summary of Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6310-5.2-1 Uncertainty Propagation for Comparison With Independent

Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6610-5.2-2 Summary: Uncertainties in Absolute Terms . . . . . . . . . . . . . . . . . . . . 6610-5.2-3 Summary of Results for Each Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

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10-5.3-1 Uncertainty Propagation for Comparative Uncertainty . . . . . . . . . . . 6810-5.3-2 Sensitivity Coefficient Estimates for Comparative Analysis . . . . . . . . 68

Nonmandatory AppendicesA Statistical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73B Uncertainty Analysis Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84C Propagation of Uncertainty Through Taylor Series . . . . . . . . . . . . . . . . . . . . . . . 87D The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

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NOTICE

All Performance Test Codes must adhere to the requirements of ASME PTC 1, GeneralInstructions. The following information is based on that document and is included here foremphasis and for the convenience of the user of the Supplement. It is expected that the Codeuser is fully cognizant of Sections 1 and 3 of ASME PTC 1 and has read them prior toapplying this Supplement.

ASME Performance Test Codes provide test procedures which yield results of the highestlevel of accuracy consistent with the best engineering knowledge and practice currentlyavailable. They were developed by balanced committees representing all concerned interestsand specify procedures, instrumentation, equipment-operating requirements, calculationmethods, and uncertainty analysis.

When tests are run in accordance with a Code, the test results themselves, withoutadjustment for uncertainty, yield the best available indication of the actual performance ofthe tested equipment. ASME Performance Test Codes do not specify means to comparethose results to contractual guarantees. Therefore, it is recommended that the parties to acommercial test agree before starting the test and preferably before signing the contract onthe method to be used for comparing the test results to the contractual guarantees. It isbeyond the scope of any Code to determine or interpret how such comparisons shall bemade.

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FOREWORD

In March 1979 the Performance Test Codes Supervisory Committee activated the PTC 19.1Committee to revise a 1969 draft of a document entitled PTC 19.1 “General Considerations.”The PTC 19.1 Committee proceeded to develop a Performance Test Code Instruments andApparatus Supplement which was published in 1985 as PTC 19.1-1985, “Measurement Uncer-tainty,” and which was intended—along with its subsequent editions—to provide a meansof eventual standardization of nomenclature, symbols, and methodology of measurementuncertainty in ASME Performance Test Codes.

Work on the revision of the original 1985 edition began in 1991. The two-fold objectivewas to improve the usefulness to the reader regarding clarity, conciseness, and technicaltreatment of the evolving subject matter, as well as harmonization with the ISO “Guide tothe Expression of Uncertainty in Measurement.” That revision was published as PTC 19.1-1998, “Test Uncertainty,” the new title reflecting the appropriate orientation of the document.

The effort to update the 1998 revision began immediately upon completion of that docu-ment. This 2005 revision is notable for the following significant departures from the 1998 text:

(a) Nomenclature adopted for this revision is more consistent with the ISO Guide. Uncer-tainties remain conceptualized as “systematic” (estimate of the effects of fixed error notobserved in the data), and “random” (estimate of the limits of the error observed from thescatter of the test data). The new aspect is that both types of uncertainty are defined at thestandard-deviation level as “standard uncertainties.” The determination of an uncertaintyat some level of confidence is based on the root-sum-square of the systematic and randomstandard uncertainties multiplied times the appropriate expansion factor for the desiredlevel of confidence (usually “2” for 95%). This same approach was used in the 1998 revisionbut the characterization of uncertainties at the standard-uncertainty level (“standard devia-tion”) was not as explicitly stated. The new nomenclature is expected to render PTC 19.1-2005 more acceptable at the international level.

(b) There is greater discussion of the determination of systematic uncertainties.(c) There is new text on a simplified approach to determine the uncertainty of straight-

line regression.ASME PTC 19.1-2005 was approved by the PTC Standards Committee on September 13,

2005, and was approved as an American National Standard by the ANSI Board of StandardsReview on November 3, 2005.

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PERFORMANCE TEST CODE COMMITTEE19.1 ON TEST UNCERTAINTY

(The following is the roster of the Committee at the time of the approval of this Supplement.)

OFFICERS

R. H. Dieck, ChairW. G. Steele, Vice ChairG. Osolsobe, Secretary

COMMITTEE PERSONNEL

J. F. Bernardin, Pratt & WhitneyD. A. Coutts, WSMSR. H. Dieck, Ron Dieck Associates, Inc.R. S. Figliola, Clemson UniversityH. K. Iyer, Colorado State UniversityJ. Maveety, Intel Corp.J. A. Rabensteine, Environmental Systems Corp.M. Soltani, Bechtel National Corp.W. G. Steele, Mississippi State University

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PERFORMANCE TEST CODES STANDARDS COMMITTEE

OFFICERS

J. G. Yost, ChairJ. R. Friedman, Vice ChairS. D. Weinman, Secretary

COMMITTEE PERSONNELP. G. Albert P. M. McHaleR. P. Allen M. P. McHaleJ. M. Burns J. W. MiltonW. C. Campbell S. P. NusplM. J. Dooley A. L. PlumleyA. J. Egli R. R. PriestleyJ. R. Friedman J. A. RabensteineG. J. Gerber J. W. SiegmundP. M. Gerhart J. A. SilvaggioT. C. Heil W. G. SteeleR. A. Johnson J. C. WestcottD. R. Keyser W. C. WoodS. J. Korellis J. G. Yost

HONORARY MEMBERSW. O. Hays F. H. Light

MEMBERS EMERITIR. L. Bannister G. H. MittendorfR. Jorgensen R. E. Sommerlad

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TEST UNCERTAINTY ASME PTC 19.1-2005

Section 1Introduction

1-1 GENERAL

This Supplement has significant additions andSections that have been rewritten to both add tothe available technology for uncertainty analysisand to make it easier for the practicing engineer.Throughout, the intent is to provide a Supplementthat can be utilized easily by engineers and scien-tists whose interest is the objective assessment ofdata quality, using test uncertainty analysis.

1-2 HARMONIZATION WITH INTERNATIONALSTANDARDS

It is recognized that this Supplement and prom-ulgated international uncertainty standards and/or guides must be in harmony. In rewriting thisSupplement, great care was taken to assure contin-ued harmony with the International Organizationfor Standardization (ISO) Guide to the Expressionof Uncertainty in Measurement (GUM) [1]. For thepracticing engineer, this harmonization means theelimination of such ambiguous terms as bias, preci-sion, bias limit, and precision index. In addition,careful attention was paid to discriminating be-tween errors, the effects of errors, and the estima-tion of their limits, which is the uncertainty.

The term “bias” is not used in this Supplement.Instead, the combined terms of “systematic error”and “systematic uncertainty” are used. The formerdescribes an error source whose effect is systematicor constant for the duration of a test. The latterdescribes the limits to which a systematic errormay be expected to go with some confidence.

The term “precision” also is not used in thisSupplement. Instead the combined terms of “ran-dom error” and “random uncertainty” are used.The former describes an error source that causesscatter in test data. The latter describes the limitsto which a random error may be expected to reachwith some confidence.

Throughout the Supplement, the term “stan-dard” uncertainty has been introduced to improve

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harmony with international guidelines and stan-dards. In this Supplement, “standard” uncertaintiesare always equivalent to a single standard devia-tion of the average.

The most common confidence level used inthis Supplement is 95% although methods foremploying alternate confidences are also given. Theconfidence level of 95% is applied to “expanded”uncertainty. This term, too, was included in thisSupplement for improved harmony with interna-tional guidelines and standards.

While this Supplement is in harmony with theISO GUM, this Supplement emphasizes the effectsof errors rather than the basis of the informationutilized in the estimation of their limits. The ISOGUM utilizes two major classifications for errorsand uncertainties. They are “Type A” and “TypeB.” Type A uncertainties have data with whichto calculate a standard deviation. Type B uncertain-ties do not have data to calculate a standarddeviation and must be estimated by other means.

This Supplement utilizes two major classifica-tions for errors and uncertainties. They are “sys-tematic” and “random.” Random errors (whoseeffects are estimated with “Random Standard Un-certainties”) cause scatter in test data. Systematicerrors (whose effects are estimated with “System-atic Standard Uncertainties”) do not.

Harmonization of this Supplement with the ISOGUM is achieved by encouraging subscripts witheach uncertainty estimate to denote the ISO Type,i.e., using subscripts of either “A” or “B.”

1-3 APPLICATIONS

This Supplement is intended to serve as a refer-ence to the various other ASME Instruments andApparatus Supplements (PTC 19 Series) and toASME Performance Test Codes and Standards ingeneral. In addition, it is applicable for all knownmeasurement and test uncertainty analyses.

The paramater values and uncertainty levelsused throughout the examples are for illustrativepurposes only and are not intended to be typicalof standard tests.



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ASME PTC 19.1-2005 TEST UNCERTAINTY

Section 2Object and Scope

2-1 OBJECT

The object of this Supplement is to define, de-scribe, and illustrate the various terms and meth-ods used to provide meaningful estimates of theuncertainty in test parameters and methods, andthe effects of those uncertainties on derived testresults.

Analysis of test measurement and result uncer-tainty is useful because it

(a) facilitates communication regarding measure-ment and test results;

(b) fosters an understanding of potential errorsources in a measurement system and the effects ofthose potential error sources on test results;

(c) guides the decision-making process for select-ing appropriate and cost-effective measurement sys-tems and methodologies;

(d) reduces the risk of making erroneous deci-sions; and

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(e) documents uncertainty for assessing compli-ance with agreements.

2-2 SCOPE

The scope of this Supplement is to specify proce-dures for evaluation of uncertainties in test parame-ters and methods, and for propagation of thoseuncertainties into the uncertainty of a test result.Depending on the application, uncertainty sourcesmay be classified either by the presumed effect(systematic or random) on the measurement ortest result, or by the process in which they maybe quantified (Type A or Type B). The variousstatistical terms involved are defined in the No-menclature (subsection 3-1) or Glossary (subsection3-2).

The end result of an uncertainty analysis is anumerical estimate of the test uncertainty with anappropriate confidence level.



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Section 3Nomenclature and Glossary

3-1 NOMENCLATURE

bXp systematic standard uncertainty com-ponent of a parameter

bXkp systematic standard uncertainty asso-

ciated with the kth elemental errorsource

bRp systematic standard uncertainty com-ponent of a result

bXYp covariance of the systematic errorsin X and Y

b+, b−p upper and lower values of nonsym-metrical systematic standard uncer-tainty

Np number of measurements or samplepoints or observations available(sample size)

Rp resultsRp random standard uncertainty of a

resultsXp standard deviation of a data sample;

estimate of the standard deviation ofthe population �x

sXp random standard uncertainty of themean of N measurements

SEEp standard error of estimate of a least-squares regression or curve fit

tp Student’s t value at a specified confi-dence level with � degrees of free-dom, i.e., t95, �

up combined standard uncertaintyUp expanded uncertainty

U+, U−p upper and lower values of the non-symmetrical expanded uncertainty

Xp individual observation in a data sam-ple of a parameter

Xp sample mean; average of a set of Nindividual observations of a pa-rameter

�p (unknown) true systematic error;fixed or constant component of �

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�p (unknown) total error; difference be-tween the assigned value of a param-eter or a test result and the truevalue

�p (unknown) true random error; ran-dom component of �

�p absolute sensitivity� ′p relative sensitivity�p (unknown) true average of a popu-

lation�p number of degrees of freedom�p (unknown) true standard deviation

of a population� 2p (unknown) true variance of a popu-

lationIndices

Ip total number of variablesip counter for variablesjp counter for individual measurements

Kp total number of sources of elementalerrors and uncertainties

kp counter for sources of elemental er-rors and uncertainties

Lp total number of correlated sourcesof systematic error

lp counter for correlated sources of sys-tematic error

Mp total number of multiple resultsmp counter for multiple resultsNp total number of measurements

3-2 GLOSSARY

calibration hierarchy: the chain of calibrations thatlinks or traces a measuring instrument to a primarystandard.

calibration: the process of comparing the responseof an instrument to a standard instrument oversome measurement range.

confidence level: the probability that the true valuefalls within the specified limits.



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degrees of freedom (�): the number of independentobservations used to calculate a standard deviation.

elemental random error source: an identifiable sourceof random error that is a subcomponent of totalrandom error.

elemental random standard uncertainty (sXk): an esti-

mate of the standard deviation of the mean of anelemental random error source.

elemental systematic error source: an identifiablesource of systematic error that is a subcomponentof the total systematic error.

elemental systematic standard uncertainty (bXk): an

estimate of standard deviation of an elementalsystematic error source.

expanded uncertainty (UX or UR): an estimate of theplus-or-minus limits of total error, with a definedlevel of confidence, (usually 95%).

influence coefficient: see sensitivity.

mean (X): the arithmetic average of N readings.

parameter: quantity that could be measured ortaken from best available information, such astemperature, pressure, stress, or specific heat, usedin determining a result. The value used is calledthe assigned value.

population mean (�): average of the set of allpopulation values of a parameter.

population standard deviation (�): a value that quanti-fies the dispersion of a population.

population: the set of all possible values of a pa-rameter.

random error (�): the portion of total error thatvaries randomly in repeated measurements of thetrue value throughout a test process.

random standard uncertainty of the sample mean(sX): a value that quantifies the dispersion of asample mean as given by eq. (4-3.3).

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result (R): a value calculated from a number ofparameters.

sample size (N): the number of individual valuesin a sample.

sample standard deviation (sx): a value that quantifiesthe dispersion of a sample of measurements asgiven by eq. (4-3.2).

sensitivity: the instantaneous rate of the changein a result due to a change in a parameter.

standard error of estimate (SEE): the measure ofdispersion of the dependent variable about a leastsquares regression or curve.

statistic: any numerical quantity derived from thesample data. X and sX are statistics.

Student’s t: a value used to estimate the uncertaintyfor a given confidence level.

systematic error (�): the portion of total error thatremains constant in repeated measurements of thetrue value throughout a test process.

systematic standard uncertainty (bX): a value thatquantifies the dispersion of a systematic errorassociated with the mean.

total error (�): the true, unknown difference betweenthe assigned value of a parameter or test resultand the true value.

traceability: see calibration hierarchy.

true value: the error-free value of a parameter ortest result.

Type A uncertainty: uncertainties are classified asType A when data is used to calculate a standarddeviation for use in estimating the uncertainty.

Type B uncertainty: uncertainties are classified asType B when data is not used to calculate astandard deviation, requiring the uncertainty tobe estimated by other methods.

uncertainty interval: an interval expressed about aparameter or test result that is expected to containthe true value with a prescribed level of confidence.



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Section 4Fundamental Concepts

4-1 ASSUMPTIONS

The assumptions inherent in test uncertaintyanalysis include the following:

(a) The test objectives are specified.(b) The test process, including the measurement

process and the data reduction process, is defined.(c) The test process, with respect to the conditions

of the item under test and the measurement systememployed for the test, is controlled for the durationof the test.

(d) The measurement system is calibrated and allappropriate calibration corrections are applied tothe resulting test data.

(e) All appropriate engineering corrections areapplied to the test data as part of the data reductionand/or results analysis process.

For expanded uncertainty, 95% confidence levelshave been used throughout this document in accor-dance with accepted practice. Other confidencelevels may be used, if required. (See Nonmanda-tory Appendix B.)

4-2 MEASUREMENT ERROR

Every measurement has error, which results ina difference between the measured value, X, andthe true value. The difference between the mea-sured value and the true value is the total error,�. Since the true value is unknown, total errorcannot be known and therefore only its expectedlimits can be estimated. Total error consists of twocomponents: random error and systematic error(see Fig. 4-2-1). Accurate measurement requiresminimizing both random and systematic errors(see Fig. 4-2-2).

4-2.1 Random Error

Random error, �, is the portion of the total errorthat varies randomly in repeated measurementsthroughout the conduct of a test. The total randomerror in a measurement is usually the sum of the

5


contributions of several elemental random errorsources. Elemental random errors may arise fromuncontrolled test conditions and nonrepeatabilitiesin the measurement system, measurement meth-ods, environmental conditions, data reduction tech-niques, etc.

4-2.2 Systematic Error

Systematic error, �, is the portion of the totalerror that remains constant in repeated measure-ments throughout the conduct of a test. The totalsystematic error in a measurement is usually thesum of the contributions of several elemental sys-tematic errors. Elemental systematic errors mayarise from imperfect calibration corrections, mea-surement methods, data reduction techniques, etc.

4-3 MEASUREMENT UNCERTAINTY

There is an inherent uncertainty in the use ofmeasurements to represent the true value. The totaluncertainty in a measurement is the combination ofuncertainty due to random error and uncertaintydue to systematic error.

4-3.1 Random Standard Uncertainty

Any single measurement of a parameter is influ-enced by several different elemental random errorsources. In successive measurements of the param-eter, the values of these elemental random errorsources change resulting in the random scatterevident in the successive measurements. If aninfinite number of measurements of a parameterwere to be taken following the defined test process,the resulting population of measurements couldbe described statistically in terms of the populationmean, �, the population standard deviation, �,and the frequency distribution of the population.These terms are illustrated in Fig. 4-3.1 for apopulation of measurements that is normally dis-tributed. For measurements with zero systematic



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Fig. 4-2-1 Illustration of Measurement Errors

error (refer to para. 4-2.2), the population meanis equal to the true value of the parameter beingmeasured and the population standard deviationis a measure of the scatter of the individual mea-surements about the population mean. For a nor-mal distribution, the interval � ± � will includeapproximately 68% of the population and the inter-val � ± 2� will include approximately 95% of thepopulation.

Since only a finite number of measurements areacquired during a test, the true population meanand population standard deviation are unknownbut can be estimated from sample statistics. Thesample mean, X, is given by

X p

�N

jp1Xj

N(4-3.1)

where Xj represents the value of each individualmeasurement in the sample and N is the number ofmeasurements in the sample. The sample standarddeviation, sX, is given by

6


sX p ��N

jp1

(Xj − X)2

N − 1(4-3.2)

Since the sample mean is only an estimate ofthe population mean, there is an inherent errorin the use of the sample mean to estimate thepopulation mean. For a defined frequency distribu-tion, the random standard uncertainty of the sam-ple mean, sX, can be used to define the probableinterval about the sample mean that is expectedto contain the population mean with a defined levelof confidence. The random standard uncertainty ofthe sample mean is related to the sample standarddeviation as follows:

sX psX

�N(4-3.3)

For a normally distributed population and alarge sample size (N > 30), the interval X ± sX isexpected to contain the true population mean with68% confidence and the interval X ± 2sX is expectedto contain the true population mean with 95%confidence [where the value 2 represents the Stu-dent’s t value for 95% confidence and degrees of



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Fig. 4-2-2 Measurement Error Components

freedom of greater than or equal to 30 wherethe degrees freedom for the random standarduncertainty is N−1 (see subsection 6-1)].

In general, increasing the number of measure-ments collected during a test and used in thepreceding formulas is beneficial as

(a) it improves the sample mean as an estimatorof the true population mean;

(b) it improves the sample standard deviation asan estimator of the true population standard devia-tion; and

(c) it typically reduces the value of the randomstandard uncertainty of the sample mean.

4-3.2 Systematic Standard Uncertainty

Every measurement of a parameter is influencedby several different elemental systematic errorsources. Each of these elemental systematic error

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sources contributes a constant, but unknown, er-ror, �Xk

, to the successive measurements of aparameter for the duration of the test (the subscriptk is used to denote a specific elemental errorsource). As �Xk

is constant for the test, the errorimparted to the average value of successive mea-surements, X [as given by eq. 4-3.1], is equivalentto the error imparted to each individual measure-ment. While �Xk

is unknown, it may be postulatedto come from a population of possible error valuesfrom which a single sample (error value) is drawnand imparted to the average measurement for thetest. Knowledge of the frequency distribution andstandard deviation of this population permits de-scribing the uncertainty in X due to this singlesample elemental systematic error in terms ofa confidence interval. The elemental systematicstandard uncertainty, bXk

, is defined as a value



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Fig. 4-3.1 Distribution of Measured Values (Normal Distribution)

that quantifies the dispersion of the population ofpossible �Xk

values at the standard deviation level.All of the elemental systematic errors associated

with a measurement combine to yield the totalsystematic error in the measurement, �X. As withelemental systematic error, total systematic erroris constant, unknown, and may be postulated tocome from a population of possible error valuesfrom which a single sample (error value) is drawnand imparted to the average measurement for thetest. Total systematic standard uncertainty, bX, isdefined as a value that quantifies the dispersionof the population of possible �X values at thestandard deviation level. Typically, total systematicstandard uncertainty is quantified by

(a) identifying all elemental sources of systematicerror for the measurement;

(b) evaluating elemental systematic standard un-certainties as the standard deviations of the possiblesystematic error distributions; and

(c) combining the elemental systematic standarduncertainties into an estimate of the total systematicstandard uncertainty for the average measurement.

4-3.2.1 Identifying Elemental Sources of System-atic Error. Attempting to identify all of the elemen-tal sources of systematic error for a measurementis an important step of an uncertainty analysis,as failure to identify any significant source ofsystematic error will lead to an underestimation

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of test uncertainty. Attempting to identify all ele-mental sources of systematic error requires a thor-ough understanding of the test objectives and testprocess. For further discussion refer to subsection5-4.

4-3.2.2 Evaluating Elemental Systematic Stan-dard Uncertainties. Once all elemental sources ofsystematic error are identified, elemental system-atic standard uncertainties for each source areevaluated. By definition, an elemental systematicstandard uncertainty is a value that quantifies thedispersion of the population of possible �Xk

valuesat the standard deviation level. As �Xk

is bothconstant and unknown during a test, successivemeasurements of a parameter do not provide suffi-cient data for direct computation of a standarddeviation as described in para. 4-3.1. Therefore,the evaluation of an elemental systematic standarduncertainty requires that a standard deviation beevaluated from engineering judgment, publishedinformation, or special data.

4-3.2.2.1 Engineering Judgment. When nei-ther published information or special data is avail-able, it is often necessary to rely upon engineeringjudgment to quantify the dispersion of errors asso-ciated with an elemental error source. In thesesituations, it is customary to use engineering analy-ses and experience to estimate the limits of the



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elemental systematic error at 95% confidence. Inother words, an interval is estimated which isexpected to contain 95% of the population ofpossible �Xk

values. Not withstanding informationto the contrary, the analyst typically assumes thatthe population of possible �Xk

values is normallydistributed, that the estimation of the limits of theerror is based upon large degrees of freedom, andthat the limits of error are symmetric (equallyspread in both the positive and negative direc-tions). Based upon these assumptions, the elemen-tal systematic standard uncertainty is estimatedas follows:

bXkp

BXk

2(4-3.4)

The variable BXkin the preceding equation repre-

sents the 95% confidence level estimate of thesymmetric limits of error associated with the kth

elemental error source. In certain situations, knowl-edge of the physics of the measurement systemwill lead the analyst to believe that the limits oferror are nonsymmetric (likely to be larger in eitherthe positive or negative direction). For treatment ofnonsymmetric systematic uncertainty see subsec-tion 8-2. The value of 2 in the equation is basedon the assumption that the population of possiblesystematic errors is normally distributed. If theanalyst thinks that the error distribution might beother than normal, such as uniform (rectangular),then a different factor would be used to convertthe 95% confidence level estimate of the systematicerror limits to an elemental systematic standarduncertainty (see Nonmandatory Appendix B). Also,there is some level of uncertainty associated withthe estimate of BXk

. This uncertainty in the estimatecan be converted into a degrees of freedom forthe systematic standard uncertainty as shown inNonmandatory Appendix B. Usually, the BXk

esti-mates are made such that this degrees of freedomwill be large (≥30). Using the recommendationsin Nonmandatory Appendix B, it can be shown thatthis large degrees of freedom (≥30) corresponds toan uncertainty in the estimate of BXk

of 13% or less.

4-3.2.2.2 Published Information. For some el-emental systematic error sources, published infor-mation from calibration reports, instrument specifi-cations, and other technical references may providequantitative information regarding the dispersionof errors for an elemental systematic error sourcein terms of a confidence interval, an ISO expanded

9


uncertainty statement, or a multiple of a standarddeviation. If the published information is presentedas a confidence interval (limits of error at a definedlevel of confidence), then the elemental systematicstandard uncertainty is estimated as the confidenceinterval divided by a statistic that is appropriatefor the frequency distribution of the error popula-tion. The specific value of this statistic must beselected on the basis of the defined confidencelevel and degrees of freedom associated with theconfidence interval. For a normal distribution, theStudent’s t statistic is used. For a 95% confidencelevel and large degrees of freedom, the value ofthe Student’s t statistic is approximated as 2 andeq. (4-3.4) would apply (refer to NonmandatoryAppendix B for values of the Student’s t statisticat other confidence levels and degrees of freedom).For situations in which the frequency distributionand degrees of freedom are unspecified, a normaldistribution and large degrees of freedom are oftenassumed. For situations involving other frequencydistributions, refer to an appropriate statistics text-book. If the published information is presented asan ISO expanded uncertainty at a defined coveragefactor (sometimes referred to as a “k factor”), thenthe elemental systematic standard uncertainty isestimated as the expanded uncertainty divided bythe coverage factor. If the published informationis presented as a multiple of a standard deviation,then the elemental systematic standard uncertaintyis estimated as the multiple of the standard devia-tion divided by the multiplier.

4-3.2.2.3 Special Data. For some elementalsystematic error sources, special data may be ob-tained that manifests the dispersion of the popula-tion of possible, unknown �Xk

values. Possiblesources of this special data include

(a) interlaboratory or interfacility tests; and(b) comparisons of independent measurements

that depend on different principles or that have beenmade by independently calibrated instruments; forexample, in a gas turbine test, airflow can be mea-sured with an orifice or a bell mouth nozzle, orcomputed from compressor speed-flow rig data,turbine flow parameters, or jet nozzle calibrations.

For these cases, the elemental systematic stan-dard uncertainty may be evaluated as follows:

bXkp � 1

N�Xk

�NX

k

j p 1

(Xkj− Xk)2

NXk− 1

(4-3.5)



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whereNXk

p the number of special data values usedin the computation of bXk

N�Xk

p the number of independent samples fromthe population of possible �Xk

values thatare averaged together in the computationof the average measurement for the test(X)

Xkp the average of the set of special data

Xkjp the jth data point of the set of special

data that manifests the dispersion of thepopulation of possible �Xk

values associ-ated with the kth elemental error source

For most measurements (especially those madeusing a single instrument calibrated at a singlelaboratory and installed in a single location), onlya single sample from the population of possible�Xk

values is included in the computation ofthe average measurement for the test (X) andhence N�X

k

p 1. The following illustrate somepossible cases where N�X

k

may be greater than one.(a) Several independent measurement meth-

ods that depend on different principles are used tomeasure the same parameter. The results from eachof the measurement methods (each determined asan average value over the duration of the test) areused as input to eq. (4-3.5) to evaluate the elementalsystematic standard uncertainty associated with theerror inherent to the various measurement methods.If the average measurement reported for the test isthe average of the results from all of the measure-ment methods, then the value for N�X

k

used in eq.(4-3.5) is equal to the number of independent mea-surement methods employed.

(b) An instrument is sent to multiple labora-tories to obtain calibration data for the instrumentprior to using the instrument in a test. The resultsfrom each of the independent laboratories (each de-termined as an offset to be applied to the instrumentwhen measuring a specific input level) are used asinput to eq. (4-3.5) to evaluate the elemental system-atic standard uncertainty associated with the errorinherent to the various laboratories. If the averagemeasurement from the instrument reported for thetest is based upon application of the average offsetfrom all of the laboratories, then the value forN�X

k

used in eq. (4-3.5) is equal to the number ofindependent laboratories employed.

4-3.2.3 Combining Elemental Systematic Stan-dard Uncertainties. Once evaluated, all of the ele-mental systematic standard uncertainties influenc-ing a measurement are combined into an estimate

10


of the total systematic standard uncertainty forthe measurement, bX. Provided all elemental sys-tematic standard uncertainties are evaluated interms of their influence on the parameter beingmeasured and in the units of the parameter beingmeasured, these elemental systematic standard un-certainties are combined per subsection 6-2. Other-wise, these elemental systematic standard uncer-tainties are combined per subsection 7-4. In somecases, elemental systematic standard uncertaintiesmay arise from the same elemental error sourceand are therefore correlated. See subsection 8-1for a detailed discussion.

4-3.3 Combined Standard Uncertainty andExpanded Uncertainty

As mentioned previously, the total uncertaintyin a measurement is the combination of uncertaintydue to random error and uncertainty due to sys-tematic error. The combined standard uncertaintyof the measurement mean, which is the total uncer-tainty at the standard deviation level, is calculatedas follows:

uX p �(bX)2 + (sX)2 (4-3.6)

where

bXp the systematic standard uncertaintysXp the random standard uncertainty of the

mean

The expanded uncertainty of the measurementmean is the total uncertainty at a defined levelof confidence. For applications in which a 95%confidence level is appropriate, the expanded un-certainty is calculated as follows:

UX p 2uX (4-3.7)

where the assumptions required for this simpleequation are presented in subsection 6-4. Expandeduncertainty is used to establish a confidence inter-val about the measurement mean which is expectedto contain the true value. Thus, the intervalX ± UX is expected to contain the true value with95% confidence (see Fig. 4-3.3).



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Fig. 4-3.3 Uncertainty Interval

4-4 PRETEST AND POSTTEST UNCERTAINTYANALYSES

(a) The objective of a pretest analysis is to estab-lish the expected uncertainty interval for a test re-sult, prior to the conduct of a test. A pretest uncer-tainty analysis is based on data and information thatexist before the test, such as calibration histories,previous tests with similar instrumentation, priormeasurement uncertainty analyses, expert opinions,and, if necessary, special tests.

A pretest uncertainty analysis allows correctiveaction to be taken, prior to expending resourcesto conduct a test, either to decrease the expecteduncertainty to a level consistent with the overallobjectives of the test or to reduce the cost of thetest while still attaining the objectives. Possiblecorrective actions include

(1) selecting alternative testing methods thatrely upon different analysis procedures, testing un-der different conditions, and/or measurement ofdifferent parameters;

11


(2) selecting alternative measurement methodsby varying test instrumentation, calibration tech-niques, installation methods, and/or measurementlocations; and

(3) increasing sample sizes by increasing sam-pling frequencies, increasing test duration, and/orconducting repeated testing.

Additionally, a pretest uncertainty analysis facili-tates communication between all parties to the testabout the expected quality of the test. This canbe essential to establishing agreement on any devi-ations from applicable test code requirements andcan help reduce the risk that disagreements regard-ing the testing method will surface after conductingthe test.

(b) The objective of a posttest analysis is to estab-lish the uncertainty interval for a test result, afterconducting a test. In addition to the data and infor-mation used to conduct the pretest uncertainty anal-ysis, a posttest uncertainty analysis is based uponthe additional data and information gathered forthe test including all test measurements, pretest and



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posttest instrument calibration data, etc. A posttestuncertainty analysis serves to

(1) validate the quality of the test result bydemonstrating compliance with test requirements;

12


(2) facilitate communication of the quality ofthe test result to all parties to the test; and

(3) facilitate interpretation of the quality of thetest by those using the test result.



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Section 5Defining the Measurement Process

5-1 OVERVIEW

The first step in a measurement uncertaintyanalysis is to clearly define the basic measurementprocess. This simple step, often overlooked, isessential to successfully develop and apply theuncertainty information. Consideration must begiven to the selection of the appropriate “truevalue” of the measurement and the time intervalfor classifying errors as systematic or random.This section provides an overview of how themeasurement process should be defined.

5-2 SELECTION OF THE APPROPRIATE “TRUEVALUE”

Depending on the user’s perspective, severalmeasurement objectives or goals and hence corres-ponding “true values” (measurements with idealzero error) may exist simultaneously in a measure-ment process. For example, when analyzing athermocouple measurement in a gas stream, sev-eral starting points or “true values” can be selected.The starting point for the analysis could beginwith the “true value” defined as the metal tempera-ture of the thermocouple junction, the gas stagna-tion temperature or junction temperature correctedfor probe effects, or the mass flow weighted aver-age of the gas temperature at the plane of theinstrumentation. Any of the aforementioned “truevalues” may be appropriate. The selection of the“true value” for the uncertainty analysis must beconsistent with the goal of the measurement [3].

5-3 IDENTIFICATION OF ERROR SOURCES

Once the true value has been defined, the errorsassociated with measuring the true value mustbe identified. Examples of error sources includeimperfect calibration corrections, uncontrolled testconditions, measurement methods, environmentalconditions, and data reduction techniques. Esti-mates to reflect the extent of these errors are

13


represented as uncertainties. These uncertaintiesin the measurement process can be grouped bysource

(a) calibration uncertainty(b) uncertainty due to test article and/or instru-

mentation installation(c) data acquisition uncertainty(d) data reduction uncertainty(e) uncertainty due to methods and other effects

5-3.1 Calibration Uncertainty

Each measurement instrument may introducerandom and systematic uncertainties. The mainpurpose of the calibration process is to eliminatelarge, known systematic errors and thus reducethe measurement uncertainty to some “acceptable”level. Having decided on the “acceptable” level,the calibration process achieves that goal by ex-changing the large systematic uncertainty of anuncalibrated or poorly calibrated instrument forthe smaller combination of systematic uncertaintiesof the standard instrument and the random uncer-tainties of the comparison. Calibrations are alsoused to provide traceability to known referencestandards or physical constants, or both. Require-ments of military and commercial contracts haveled to the establishment of extensive hierarchiesof standards laboratories. In some countries, anational standards laboratory is at the apex ofthese hierarchies, providing the ultimate referencefor every standards laboratory. Each additionallevel in the calibration hierarchy adds uncertaintyin the measurement process (see Fig. 5-3.1).

5-3.2 Uncertainty Due to Test Article and/orInstrumentation Installation

Measurement uncertainty can also exist frominteractions between (a) the test instrumentationand the test media or (b) between the test articleand test facility. Examples of these types of uncer-tainty are



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Fig. 5-3.1 Generic Measurement Calibration Hierarchy

(a) Interactions Between the Test Instrumentationand Test Media:

(1) Installation of sensors in the test media maycause intrusive disturbance effects. An examplecould be the measurement of airflow in an air condi-tioning duct. Depending on the design of the pitotstatic probe, it may affect the measured total andstatic pressure and thus the calculated airflow.

(2) Environmental effects on sensors/instru-mentation may exist when the sensors experienceenvironmental effects that are different from thoseobserved during calibration. These may be suchthings as conduction, convection, and radiation ona sensor when installed in a gas turbine.

(b) Interactions Between the Test Article and TestFacility:

(1) Test-facility limitations for certification test-ing affects product measurement uncertainty. Anexample may be an air conditioner that was benchtested in a laboratory but used in an automotivemechanics shop. The effect of the oily air can influ-ence the quoted rating of the unit. A second exampleis the testing of a gas turbine engine in an altitudefacility. The facility simulates altitude by loweringthe ambient pressure at the test article exhaust andraising the inlet pressure at the engine inlet. In appli-

14


cation the inlet pressure is elevated due to the ramdrag effects of the aircraft. A correction factor mustbe applied that corrects between uninstalled to in-stalled aircraft engine performance.

(2) Facility limitations for testing may requireextrapolations to other conditions. An example isthe testing of an automotive engine. The fuel con-sumption of an automotive engine changes withaltitude and speed. An automotive test facility mayonly be able to test at specified altitudes and speeds,and the effects at other altitude conditions may needto be extrapolated.

5-3.3 Data Acquisition Uncertainty

Uncertainty in data acquisition systems can arisefrom errors in the signal conditioning, the sensors,the recording devices, etc. The best method tominimize the effects of many of these uncertaintysources is to perform overall system calibrations.By comparing known input values with their mea-sured results, estimates of the data acquisitionsystem uncertainty can be obtained. However, itis not always possible to do this. In these cases,it is necessary to evaluate each of the elemental



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


uncertainties and to combine them to predict theoverall uncertainty.

5-3.4 Data Reduction Uncertainty

Computations on raw data are done to produceoutput (data) in engineering units. Typical uncer-tainty sources in this category stem from curvefits and computational resolution. With the recentadvances in computer systems, the computationalresolution uncertainty sources are often negligible;however, curve-fit error uncertainty can be signifi-cant. Other examples of data reduction uncertaintyinclude

(a) the assumptions or constants contained in thecalculation routines;

(b) using approximating engineering relation-ships or violating their assumptions; and

(c) using an empirically derived correlation suchas empirical fluid properties.

These additional uncertainties may be of eithera systematic or random nature depending on theireffect on the measurement.

5-3.5 Uncertainty Due to Methods and OtherEffects

Uncertainties due to methods are defined asthose additional uncertainty sources that originatefrom the techniques or methods inherent in themeasurement process. These uncertainty sources,beyond those contained in calibration, installationsources, data acquisition, and data reduction, maysignificantly affect the uncertainty of the finalresults.

5-4 CATEGORIZATION OF UNCERTAINTIES

This Standard delineates uncertainties by theeffect of the error (i.e., systematic and random).This categorization approach supports the identifi-cation, understanding, and managing of test uncer-tainties. If the nature of an elemental error is fixedover the duration of the defined measurementprocess, then the error contributes to the systematicuncertainty. If the error source tends to causescatter in repeated observations of the definedmeasurement process, then the source contributesto the random uncertainty.

Because measurement uncertainties are catego-rized by the effect of the error, the time intervaland duration of the measurement process can beimportant considerations and so must be clearly

15


stated. The significance of this is discussed in para.5-4.2. In addition, the objective of the test mayaffect the categorization as discussed in para. 5-4.3.

5-4.1 Alternate Categorization Approach

An alternate approach, which is used in the ISOGUM, categorizes the uncertainties based on themethod used to estimate uncertainty. Those evalu-ated with statistical methods are classified as TypeA, while those, which are evaluated by othermeans, are classified as Type B. Depending onthe selection of the defined measurement process,there may be no simple correspondence betweenrandom or systematic and Type A or Type B.

5-4.2 Time Interval Effects

Errors that may be fixed over a short time periodmay be variable over a longer time period. Forexample, calibration corrections, which are as-sumed fixed over the life of the calibration interval,can be considered variable if the process consistsof a time interval encompassing several differentcalibrations. The time interval must be clearlyspecified to classify an error, and it may not alwaysbe the same interval as the test duration. Forexample, when comparing results among variouslaboratories, it may be appropriate to classify anerror as random rather than as systematic eventhough that error may have been constant for theduration of any single test.

The effects of a time interval may also be impor-tant when considering the stability and control ofa test process. The stability of a measurementmethod is a generic concept related to the closenessof agreement between test results. Process stabilityis estimated from observations of scatter within adata set and is treated as a random error. Variabil-ity in independent test results obtained underdifferent test conditions, varying experimental set-ups, or configuration changes allow for additionalbetween-test random errors.

5-4.3 Test Objective

The classification and number of error sourcesare often affected by the test objective. For example,if the test objective is to measure the average gasmileage of model “XYZ” cars, the variabilityamong or between cars of the same model mustbe considered. Random error obtained in a testfrom a given car would not include car-to-car



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Fig. 5-4.3 Difference Between “Within” and “Between” Sources of Data Scatter

variations and thus would not represent all randomerror sources. To observe the random error associ-ated with car-to-car variability, the experimentwould need to be run again using a randomselection of different cars within the same model(see Fig. 5-4.3). The total variation in the test resultis greater than that observed from a test of asingle given car. This variation would be morerepresentative of the total random error associatedwith determining gas mileage for the fleet of model“XYZ” cars. Of course, if the data of interest isgas mileage of a given single car, then the estimatedvariation with testing the representative given caris an appropriate estimate for the random error.The same short-term and long-term effects mustbe applied for other variables affecting gas mileage(temperature, altitude, humidity, road conditions,driver variations, etc.).

16


5-5 COMPARATIVE VERSUS ABSOLUTE TESTING

The objective of a comparative test (also knownas a back-to-back test) is to determine, with thesmallest measurement uncertainty possible, the neteffect of a design change. The first test is run withthe standard or baseline configuration. The secondtest is then run in the same facility with the designchange and hopefully with instruments, setups,and calibrations identical to those used in the firsttest. The difference between the results of thesetests is an indication of the effect of the designchange. Depending on whether common instru-mentation, setups, and calibrations are used be-tween comparative tests, the effects of correlateduncertainties (see Section 8) may cause the totaluncertainty of the difference between the test re-sults to be less than the uncertainty of each separatetest result. An example of back-to-back uncertaintyanalysis is shown in Example 8.1 in subsection 8-1.



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Section 6Uncertainty of a Measurement

6-1 RANDOM STANDARD UNCERTAINTY OF THEMEAN

6-1.1 General Case

For X that is determined as the average of Nmeasurements, the appropriate random standarduncertainty of the mean (sX) is given by eq. (4-3.3). This type of estimate is an ISO Type Aestimate.

In a sample of measurements, the degrees offreedom is the sample size (N). When a statistic iscalculated from the sample, the degrees of freedomassociated with the statistic is reduced by one forevery estimated parameter used in calculating thestatistic. For example, from a sample of sizeN, X is calculated by eq. (4-3.1). The sample stan-dard deviation (sX) and the random standard un-certainty of the mean (sX) are calculated from eqs.(4-3.2) and (4-3.3), respectively, and each has N − 1degrees of freedom (�)

� p N − 1 (6-1.1)

because X (based on the same sample of data) isused in the calculation of both quantities.

6-1.2 Using Previous Values of sX

In some test situations, the measurement of avariable may be only a single measurement or anaverage of measurements taken over a short timeframe, as with a computer-based data acquisitionsystem. In this latter case, the time frame overwhich the measurements are taken may be on theorder of milliseconds or less while the randomvariations in the process may be on the order ofseconds, or minutes, or even days. This “short timeframe averaged” value should then be handled inthe same manner as a single measurement.

Information about the possible variations in asingle measurement must be obtained from previ-ous measurements of the variable taken over thetime frame and conditions that cover the variations

17


in the variable. For example, taking multiple mea-surements as a function of time while holdingall other conditions constant would identify therandom variation associated with the measurementsystem and the unsteadiness of the test condition.If the sample standard deviation of the variablebeing measured is also expected to be representa-tive of other possible random variations in themeasurement, e.g., repeatability of test conditions,variation in test configuration, etc., then theseadditional error sources will have to be variedwhile the multiple data measurements are takento determine the standard deviation.

Another situation where previous values of avariable would be useful is when a small samplesize (N) is used to calculate the mean value (X)of a measurement. If a much larger set of previousmeasurements for the same test conditions is avail-able, then it could be used to calculate a moreappropriate standard deviation for the currentmeasurement [4]. Typically, these larger data setsare taken in the early phases of an experimentalprogram. Once the random variation of the testvariables is understood, then this information canbe used to streamline the test procedure by reduc-ing the number of measurements taken in the laterphases of the test.

When NP previous values (XPj) are known forthe quantity being measured, the sample standarddeviation for the variable can be calculated as

sX p sXPp � 1

NP − 1 �N

P

jp1(XPj

− XP)2�1 ⁄2

(6-1.2)

where

XP p1

NP�

NP

jp1XPj

(6-1.3)

The appropriate random standard uncertaintyof the mean for the current measurement (X) is then



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


sX psX

�N(6-1.4)

where N is the number of current measurementsaveraged to determine X. The number of degreesof freedom for this random standard uncertaintyof the mean sX is

� p NP − 1 (6-1.5)

This estimate of the random standard uncer-tainty is an ISO Type A estimate since it is obtainedfrom data. The case where the data sample isonly a single measurement is handled above withN p 1.

6-1.3 Using Elemental Random Error Sources

Another method of estimating the random stan-dard uncertainty of the mean for a measurementis from information about the elemental randomerror sources in the entire measurement process.If all the random standard uncertainties are ex-pressed in terms of their contribution to the mea-surement, then the random standard uncertaintyfor the measurement mean is the root-sum-squareof the elemental random standard uncertainties ofthe mean from all sources divided by the squareroot of the number of current readings (N) aver-aged to determine X

sX p1

�N��

K

kp1(s

Xk

)2�1 ⁄2

(6-1.6)

where K is the total number of random error (oruncertainty) sources.

Each of the elemental random standard uncer-tainties of the mean sXk

is calculated using themethods described in para. 6-1.1 or 6-1.2 dependingupon which is appropriate, and each is assumedto be an ISO Type A estimate. If in each of theN measurements of the variable X the output ofan elemental component is averaged Nk times toobtain Xk, then the method in para. 6-1.1 wouldbe used. If instead previous information is usedto obtain sXk

, then the method in para. 6-1.2would apply.

The degrees of freedom for the estimated ran-dom standard uncertainty of the mean (sX) isdependent on the information used to determineeach of the elemental random standard uncertain-ties of the mean and is calculated as

18


� p� �

K

kp1(sXk)2�

2

�K

kp1

(sXk)4

�k

(6-1.7)

where �k is the appropriate degrees of freedomfor sXk

and is obtained from eq. (6-1.1) or (6-1.5)as appropriate. When all error sources have largesample sizes, the calculation of � is unnecessary.However, for small samples, when combining ele-mental random standard uncertainties of the meanby the root-sum-square method [see eq. (6-1.6)],the degrees of freedom (�) associated with thecombined random standard uncertainty is calcu-lated using the Welch-Satterthwaite formula [5]above [eq. (6-1.7)].

6-1.4 Using Estimates of Sample StandardDeviation

In a pretest uncertainty analysis, previous infor-mation might not be available to estimate thesample standard deviation as discussed in para.6-1.2 or 6-1.3. In this case, an estimate of thesample standard deviation (sX) would be madeusing engineering judgment and the best availableinformation. This type of uncertainty estimatewould be an ISO Type B estimate.

6-2 SYSTEMATIC STANDARD UNCERTAINTY OF AMEASUREMENT

The systematic standard uncertainty bX of ameasurement was defined in para. 4-3.2 as a valuethat quantifies the dispersion of the systematicerror associated with the mean. The true systematicerror (�) is unknown, but bX is evaluated so thatit represents an estimate of the standard deviationof the distribution for the possible � values. Itshould be noted that while bX is an estimate ofthe dispersion of the systematic errors in a mea-surement, the systematic error that is present ina specific measurement is a fixed single value of �.

The systematic standard uncertainty of the mea-surement is the root-sum-square of the elementalsystematic standard uncertainties bXk

for allsources.

bX p � �K

kp1(bXk

)2�1⁄2

(6-2.1)



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


where

Kp the total number of systematic errorsources

and eachbXk

p an estimate of the standard deviation ofthe kth elemental error source

Note that in eq. (6-2.1) all of the elementalsystematic standard uncertainties are expressed interms of their contribution to the measurement.

For each systematic error source in the measure-ment, the elemental systematic standard uncer-tainty must be estimated from the best availableinformation. Usually these estimates are made us-ing engineering judgment (and are therefore ISOType B estimates). Sometimes previous data areavailable to make estimates of uncertainties thatremain fixed during a test (and are therefore ISOType A estimates). If any of the elemental system-atic uncertainties are nonsymmetrical, then themethod given in subsection 8-2 should be usedto determine the systematic standard uncertaintyof the measurement.

There can be many sources of systematic errorin a measurement, such as the calibration process,instrument systematic errors, transducer errors,and fixed errors of method. Also, environmentaleffects, such as radiation effects in a temperaturemeasurement, can cause systematic errors ofmethod. There usually will be some elementalsystematic standard uncertainties that will be dom-inant. Because of the resulting effect of combiningthe elemental uncertainties in a root-sum-squaremanner, the larger or dominant ones will controlthe systematic uncertainty in the measurement;however, one should be very careful to identityall sources of fixed error in the measurement.

6-3 CLASSIFICATION OF UNCERTAINTY SOURCES

As discussed in subsection 1-1, the ISO Guide[1] classifies uncertainties by source as either TypeA or Type B. Type A uncertainties are the calcu-lated standard deviations obtained from data sets.Type B uncertainties are those that are estimatedor approximated rather than being calculated fromdata. Type B uncertainties are also given as stan-dard deviation level estimates.

In this Supplement, uncertainties are classifiedby their effect on the measurement, either randomor systematic, rather than by their source. Thiseffect classification is chosen since most test opera-tors are concerned with how errors in the testwill affect the measurements.

19


There may be situations when it is convenientto classify elemental uncertainties by both effectand source. Such classifications would be usefulin international test programs. This Supplementrecommends the following nomenclature for dualclassification:

bXk,Ap elemental systematic standard uncertainty

calculated from data, as in a calibrationprocess

bXk,Bp elemental systematic standard uncertainty

estimated from the best available infor-mation

sXk,Ap elemental random standard uncertainty

calculated from datasXk

,Bp elemental random standard uncertaintyestimated from best available information

6-4 COMBINED STANDARD AND EXPANDEDUNCERTAINTY OF A MEASUREMENT

For simplicity of presentation, a single value isoften preferred to express the estimate of the errorbetween the mean value (X) and the true value,with a defined level of confidence. The interval

X ± UX (6-4.1)

represents a band about X within which the truevalue is expected to lie with a given level ofconfidence (see Fig. 4-3.3). The uncertainty intervalis composed of both the systematic and randomuncertainty components.

The general form of the expression for determin-ing the uncertainty of a measurement is the root-sum-square of the systematic and random standarduncertainties of the measurement, with this quan-tity defined as the combined standard uncertainty(uX) [1].

uX p �(bX)2 + (sX)2 (6-4.2)

where

bXp the systematic standard uncertainty [eq.(6-2.1)]

sXp the random standard uncertainty of themean [eq. (4-3.3), (6-1.4), or (6-1.6) as ap-propriate]

In order to express the uncertainty at a specifiedconfidence level, the combined standard uncer-



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Table 6-4-1 Circulating Water Bath Temperature Measurements(Example 6-4.1)

Measured Measured MeasuredElapsed Temp., Elapsed Temp., Elapsed Temp.,

Time, min °C Time, min °C Time, min °C

0 85.11 11 85.28 21 85.231 84.89 12 85.11 22 85.122 85.07 13 84.80 23 85.433 84.77 14 84.79 24 84.504 85.24 15 85.22 25 85.22

5 84.72 16 85.05 26 85.396 85.00 17 84.58 27 84.747 85.39 18 85.20 28 85.358 84.72 19 85.14 29 84.759 85.50 20 85.05 30 84.5610 85.18

tainty must be multiplied by an expansion factortaken as the appropriate Student’s t value forthe required confidence level (see NonmandatoryAppendix B). Depending on the application, vari-ous confidence levels may be appropriate. TheStudent’s t is chosen on the basis of the level ofconfidence desired and the degrees of freedom.The degrees of freedom used is a combined degreesof freedom based on the separate degrees of free-dom for the random standard uncertainty and theelemental systematic standard uncertainties (seeNonmandatory Appendix B). A t value of 1.96(usually taken as 2) corresponds to large degreesof freedom and defines an interval with a level ofconfidence of approximately 95%. This expansionfactor of 2 is used for most engineering applica-tions. For other confidence levels see Nonmanda-tory Appendix B.

The uncertainty for 95% confidence and largedegrees of freedom (t p 2) is calculated by

UX p 2uX (6-4.3)

where

uX p the combined standard uncertainty [eq.(6-4.2)]

Example 6-4.1. A digital thermometer was usedto measure the average temperature of a circulatingwater bath that is being used in an experiment. Theexperiment lasted a total of 30 min. Temperaturemeasurements were collected every minute re-sulting in a total of 31 data points as presentedin Table 6-4-1.

20


(a) Uncertainty Due to Random Error. The uncer-tainty due to the random error of the average tem-perature measurement is evaluated as follows.

(1) The sample mean, or average value, of thetemperature measurements is determined using eq.(4-3.1) as follows:

X p1N �

N

jp1Xj p 85.04°C

(2) The sample standard deviation is deter-mined using eq. (4-3.2) as follows:

sX p ��N

jp1

(Xj − X)2

N − 1p 0.28°C

(3) The random standard uncertainty of thesample mean is determined using eq. (4-3.3) asfollows:

sX psX

�Np 0.05°C

(b) Uncertainty Due to Systematic Error. The uncer-tainty due to the systematic error of the averagecirculating water bath temperature measurement isevaluated by

(1) identifying all elemental sources of system-atic error for the measurement;



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Table 6-4-2 Systematic Uncertainty of Average Circulating Water BathTemperature Measurements (Example 6-4.1)

Description of Systematic Uncertainty Source Systematic Standard Uncertainty, °C

Calibration of digital thermometer 0.05Environmental influences (ambient 0.005

temperature, humidity, etc.) on digitalthermometer

Effects of conduction and radiation heat 0.0005transfer between the digital thermometerand the surroundings

Uniformity of circulating water bath (spatial 0.05uncertainty)

Total systematic uncertainty 0.07

(2) evaluating elemental systematic standarduncertainties as the standard deviations of the possi-ble systematic error distributions; and

(3) combining the elemental systematic stan-dard uncertainties into an estimate of the total sys-tematic standard uncertainty for the measurement.

For the purposes of this example, a summaryof this evaluation is presented in Table 6-4-2. Referto para. 4-3.2 and subsections 6-2, 7-4, 8-1, and 8-2for further discussion of the process for identifying,evaluating, and combining elemental systematicuncertainties.

21


(c) Expanded Uncertainty. The expanded uncer-tainty of the average circulating water bath tempera-ture measurement is evaluated using eqs. (6-4.2) and(6-4.3) as follows:

UX p 2�(bX)2 + (sX)2 p 0.17°C

Therefore, the true average temperature of thecirculating water during the experiment is expectedto lie within the following interval with 95% confi-dence:

X ± UX p 85.04 ± 0.17°C



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Section 7Uncertainty of a Result

7-1 PROPAGATION OF MEASUREMENTUNCERTAINTIES INTO A RESULT

Calculated results, such as in an estimate ofefficiency, are not usually measured directly. In-stead, more basic parameters, such as temperatureand pressure, are either measured or assigned andthe required result is calculated as a functionof these parameters, including using unmeasuredproperties, such as tabulated coefficients. Uncer-tainties in these measurements or assigned valuesof the parameters are propagated to the resultthrough the functional relationship between theresult and the parameters. The effect of the propa-gation can be approximated by the Taylor seriesmethod (see Nonmandatory Appendix C).

To estimate the uncertainty of a calculated result,the result, R, is expressed in terms of the averageor assigned values of the independent parameters(Xi) that enter into the result. That is,

R p f(X1, X2, . . .,XI) (7-1.1)

where the subscript I signifies the total numberof parameters involved in R, and the averagevalues of the independent parameters are ob-tained as

Xi p1Ni

�Ni

jp1Xij

(7-1.2)

where Ni is the number of repeated measurementsof Xi. Ni will be 1 for a single data point orassigned value of a parameter.

7-1.1 Single and Multiple Tests

In some experimental situations, a set of parame-ters (Xi) is measured and a single result, R, iscalculated. This case is called a single test result.In this case, some of the parameters may be basedon single measurements and others may be the

22


mean values based on Ni repetitions. Ni can bedifferent for each Xi.

Repeated tests are those run under the sameconditions to estimate a parameter or a set ofparameters (Xi). The statistics from repeated testsallow for quantifying the expected variation in aparameter or in a result derived from parameters.The estimation of the sample mean and standarddeviation based on Ni measurements is calculatedas described in para. 4-3.1.

When tests are duplicated under similar butsomehow changed operating conditions, these gen-erate multiple data sets for the measured parame-ters. The statistics found by combining these dupli-cated data sets allow for a reasonable estimate ofthe variations possible in the result that might bedue to the control of test operating conditions, oruse of different test rigs, instrumentation, or testlocation. Whereas these influences might normallybe considered systematic errors during repeatedtests, the duplicated tests can randomize thesesystematic errors providing error estimates fromthe statistical variations in the combined data pool[6]. The overall reported result will usually becombined to provide the mean of the multipleresults, R.

Careful consideration should be given to design-ing the test series to average as many causes ofvariation as possible within cost constraints. Thetest design should be tailored to the specific situa-tion. For example, if experience indicates that time-to-time and test apparatus-to-apparatus variationsare significant, a test design that averages multipletest results on one rig or for only one day mayproduce optimistic random uncertainty estimatescompared to testing several rigs, each monitoredseveral times over a period of several days. Thelist of test variation causes are many and mayinclude the above plus environmental and testcrew variations. Historic data are invaluable forstudying these effects. A statistical technique calledanalysis of variance (ANOVA) is useful for parti-tioning the total variance by source. If the pretest



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


uncertainty analysis (see subsection 4-4) identifiesunacceptably large error sources, special tests tomeasure the effects of these sources should beconsidered.

7-2 SENSITIVITY

Sensitivity is the instantaneous rate of changein a result to a change in a parameter. Twoapproaches to estimating the sensitivity coefficientof a parameter are discussed in the following text.

7-2.1 Analytically

When there is a known mathematical relation-ship between the result, R, and its parameters(X1, X2, . . ., XI), then the absolute (dimensional)sensitivity coefficient (�i) of the parameter Xi maybe obtained by partial differentiation.

Thus, if R p f(X1, X2, . . ., XI), then

�i p∂R

∂Xi(7-2.1)

Analogously, the relative (nondimensional) sensi-tivity coefficient (�i′) is

�i′ p

∂RR

∂Xi

Xi

pXi

R � ∂R

∂Xi� (7-2.2)

7-2.2 Numerically

Finite increments in a parameter also may beused to evaluate sensitivity using the data reduc-tion calculation procedure. In this case, �i isgiven by

�i′ p�R

�Xi

(7-2.3)

and �i′ by

�i′ p

�RR

�Xi

Xi

pXi

R � �R

�Xi� (7-2.4)

The result is calculated using Xi to obtain R [7].The derivatives and values for �R are best esti-mated by use of central differencing methods.

23


Numerical differentiation is covered in variousreferences [e.g., 8].

To best approximate the sensitivity that wouldbe obtained analytically, the value of �Xi usedshould be as small as practical (i.e., large enoughto keep truncation errors from influencing thecalculations).

7-3 RANDOM STANDARD UNCERTAINTY OF ARESULT

7-3.1 Single Test

The absolute random standard uncertainty of asingle test result may be determined from thepropagation equation (see Nonmandatory Appen-dix C) as

sR p ��I

ipi��i sXi

�2�1⁄2

(7-3.1)

The relative random standard uncertainty of aresult is

sR

Rp ��I

ipi �� ′isXi

Xi�

2�1⁄2

(7-3.2)

The symbols �i and �i′ are the absolute and relativesensitivity coefficients, respectively, of eqs. (7-2.1)or (7-2.3) and (7-2.2) or (7-2.4), and sXi

is therandom standard uncertainty of the measured pa-rameter average (Xi), determined according to themethods presented in subsection 6-1.

7-3.2 Multiple Tests: Repeated Tests

When more than one test is conducted with thesame instrument package (i.e., repeated tests), theuncertainty of the average test result may be re-duced from that for one test because of the reduc-tion in the random uncertainty of the average.However, systematic uncertainty will remain thesame as for a single test.

The average result from more than one test isgiven by

R p

�M

mp1Rm

M(7-3.3)



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


where M signifies the number of tests available.Following eq. (4-3.1), the estimate of the standarddeviation of the distribution of the results is

sR p � �M

mp1(Rm − R)2

M − 1�

1 ⁄2

(7-3.4)

where sR includes random errors within tests andvariation between tests. The degrees of freedomassociated with sR is determined by � p M − 1.The random standard uncertainty of the resultis estimated directly from the sample standarddeviation of the mean result from multiple testsand is

sR psR

M(7-3.5)

This random standard uncertainty of the resultalso has �R pM − 1 degrees of freedom.

7-3.3 Multiple Tests: Combined Tests

When tests are duplicated under similar butsomehow changed operating conditions, the aver-age result and average standard deviations aregiven by combining the information of each test.The statistical equations used to assess variationsbetween tests are similar to those discussed inpara. 7-3.2 but discussion of their interpretationis beyond the intention of this document and ispresented in detail in reference [6].

7-4 SYSTEMATIC STANDARD UNCERTAINTY OF ARESULT

The absolute systematic standard uncertainty ofa result may be determined from the propagationequation (see Nonmandatory Appendix C) as

bR p � �I

ip1(�i bXi

)2�1⁄2

(7-4.1)

The relative systematic standard uncertainty ofa result is

bR

Rp � �

I

ip1 ��i′bXi

Xi�

2

�1⁄2

(7-4.2)

The symbol bXiis the systematic standard uncer-

tainty of the measured parameter (see subsection

24


6-2). Equations 7-4.1 and 7-4.2 assume the system-atic uncertainties for the different parameters areindependent.

7-5 COMBINED STANDARD UNCERTAINTY ANDEXPANDED UNCERTAINTY OF A RESULT

The general form of the expression for determin-ing the combined standard uncertainty of a resultis the root-sum-square of both the systematic andthe random standard uncertainty of the result.The following simple expression for the combinedstandard uncertainty of a result applies in manycases:

uR p (bR)2 + (SR)2�1 ⁄2

(7-5.1)

where bR is obtained from eq. (7-4.1) and sR isobtained from either eq. (7-3.1) for a single testresult or from eq. (7-3.5) for a multiple test result.

The expanded uncertainty in the result at ap-proximately 95% confidence is given by

UR,95 p 2uR (7-5.2)

where the use of the factor of 2 assumes sufficientlylarge degrees of freedom for the 95% confidencelevel (i.e., t95 p 2). This factor can be modified asappropriate for other confidence levels and smalldegrees of freedom as discussed in NonmandatoryAppendix B. The interval within which the trueresult should lie with 95% confidence is given as

R ± UR,95 (7-5.3)

The special cases of correlated systematic uncer-tainties and nonsymmetric systematic uncertaintiesare covered in subsections 8-1 and 8-2, respectively.Treatment of uncertainty intervals with alternateconfidence levels, error distributions, and alternateuncertainty equations is addressed in Nonmanda-tory Appendix B.

With high-speed computing capabilities, MonteCarlo methods have become popular for determin-ing the test result uncertainty using the test inputvariable values and their associated uncertainties.

7-6 EXAMPLES OF UNCERTAINTY PROPAGATION

7-6.1 The Magnitude of Uncertainty in the Test

The magnitude of uncertainty in a test can bequantified by using the data reduction equations



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Table 7-6.1-1 Table of Data (Example 7-6.1)

Independent Parameters

UncertaintyContribution of

Parameters to theResult

Parameter Information (in Result Units(in Parameter Units) Squared)

AbsoluteAbsolute Absolute Systematic Absolute

Systematic Random Standard RandomStandard Standard Absolute Uncertainty Uncertainty

Nominal Standard Uncertainty, Uncertainty, Sensitivity, Contribution, Contribution,Symbol Description Units Value Deviation Ni bX

iSX

i�i �ibX

i

2 �isXi

2

P Power W 18.4 0.65 50 0.1 0.092 0.163 2.67E-04 2.25E-04L Length m 0.025 0.0012 32 6.4E-05 2.12E-04 120.3 5.9E-05 6.50E-04A Cross- m2 0.018 0.0013 32 6.4E-05 2.30E-04 −167 1.14E-04 1.47E-03

sectional area�T Temperature °C 8.5 0.5 50 0.1 0.071 −0.354 1.25E-03 6.26E-04

difference

and estimates for each of the test parameters. Thisis important and useful because it provides insightinto whether a particular test or methodology isfeasible based on the acceptable order of magnitudein uncertainty. If for example the uncertainty inthe results needs to be 10% or less, and the pretestcalculations show the best results will be on theorder of 15%, then appropriate corrective actionto reduce the uncertainty can be taken.

Example 7-6.1: Tests are often conducted usingthe guarded hot plate technique to determine thethermal conductivity of a material. The guardedhot plate is used because it is relatively inexpensiveand effective in providing the boundary conditionsnecessary to ensure one dimensional heat flowthrough the material. For steady state conditions,the one-dimensional form of Fourier’s law can beused to characterize the thermal conductivity ofthe test specimen.

k pPA

L�T

(7-6.1)

The result, R, which in this example is given bythe thermal conductivity k, is determined as afunction of the parameters using their nominalvalues and eq. (7-6.1), which itself is just thespecific form of eq. (7-1.1). In eq. (7-6.1), P is theelectrical power dissipated by the hot plate, �Tis the temperature difference across the hot and

25


cold surfaces, A is the cross-sectional area, and Lis the thickness of the material, respectively. Theuncertainty in conductivity will be directly depen-dent on the method used to measure each parame-ter in the computation. To estimate the expectedconductivity uncertainty, we must know or esti-mate the measurement errors associated with thepower, temperature, and geometry. Table 7-6.1-1lists the independent parameters for this exampleand Table 7-6.1-2 the final results.

From the previous analysis, the conductivity wasdetermined to be k p 3.01 ± 0.14 W/mK at 95%confidence. Inspection of Table 7-6.1-1 lends insightinto the magnitude of the contributions to theoverall uncertainty. The uncertainty in power re-quired to generate a constant heat flux will dependon the type of material being tested. Therefore, itis important to select the power setting such thatmeasurements are at least at the midscale of thepower supply range.

7-6.2 Ranking of Uncertainty Components

Uncertainty analysis can be used to identify thelargest contributors to the overall uncertainty. Thisenables designers and engineers to make pretesthardware and/or experimental methodology im-provements.

Example 7-6.2: A test is to be conducted todetermine the pressure loss coefficient across a



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Table 7-6.1-2 Summary of Data (Example 7-6.1)

Absolute Absolute ExpandedSystematic Random Combined Uncertainty

Calculated Standard Standard Standard of theValue, Uncertainty, Uncertainty, Uncertainty, Result,

Symbol Description Units R bR sR uR UR, 95

K Conductivity W/mK 3.01 0.041 0.055 0.068 0.14

Table 7-6.2-1 Table of Data (Example 7-6.2)

Parameter Information

Uncertainty Contribution ofParameter Information Parameters to the Result

(in Parameter Units) (in Result Units Squared)

Absolute AbsoluteAbsolute Absolute Systematic Random

Systematic Random Standard StandardStandard Standard Absolute Uncertainty Uncertainty

Nominal Standard Uncertainty, Uncertainty, Sensitivity, Contribution, Contribution,Symbol Description Units Value Deviation Ni bX

iSX

i�i �ibX

i

2 �iSXi

2

�P Pressure Pa 230.0 10.0 50 5.0 1.41 0.0062 9.70E-04 7.64E-05drop

D Diameter m 0.030 1.27E-04 32 6.35E-06 2.25E-05 191.0 1.47E-06 1.85E-05Q Volumetric m3/s 0.0116 1.20E-03 32 1.0E-04 2.12E-04 −247.0 6.10E-04 2.74E-03

flow rate� Air density kg/m3 1.192 . . . . . . 0.013 . . . −1.202 2.45E-04 . . .

valve at a specified operating condition. The appa-ratus consists of a closed loop tube containinga differential pressure gauge for measuring thepressure drop across the valve, a blower for cyclingthe working fluid (air), and a flow meter. Theequation characterizing the loss coefficient is

K p�P

12

�V2(7-6.2)

where �P is the pressure drop across the valve,� is the density of air and V is the average airvelocity. From the definition of mass flow rate,the velocity can be replaced with the volumetricflowrate Q through the expression Q p AV, whereA is the cross-sectional area of the tube. In thisexample we will consider a round tube so A p�D2/4. Substituting into eq. (7-6.2) gives

K p�2

8D4�P

�Q2 (7-6.3)

26


The Mach number for this flow is well below0.3, and the density is assumed to follow the idealgas equation of state, � p P/RT. Measurements ofthe ambient air were made, and the data aregiven as:

P p 101.3 kPa sP p 8.0 kPa NP p 50T p 296.2 K sT p 1.6 K NT p 50

Following the procedure outlined in subsection7-1, the density and expanded uncertainty in den-sity were determined to be � p 1.192 ± 0.027 kg/m3 at 95% confidence, where the uncertainty contri-bution from the gas constant was assumed negligi-ble (9). Data from the loss coefficient test aresummarized in Tables 7-6.2-1 and 7-6.2-2. Thedensity uncertainty is treated as simply a system-atic contribution in the loss coefficient calculation.

From the previous analysis, the loss coefficientwas determined to be K p 1.433 ± 0.137 at 95%



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Table 7-6.2-2 Summary of Data (Example 7-6.2)

Calculated Results

CombinedAbsolute Absolute Standard Expanded

Systematic Random Uncertainty UncertaintyCalculated Standard Standard of the of the

Value, Uncertainty, Uncertainty, Result, Result,Symbol Description Units R bR sR uR UR, 95

K Loss Dimensionless 1.433 0.043 0.053 0.068 0.137coefficient

Fig. 7-6.2 Pareto Chart of Systematic and Random Uncertainty Component Contributions to CombinedStandard Uncertainty

confidence. A pareto chart (see subsection A-4) ofthe systematic and random uncertainty contribu-tions to the loss coefficient is given in Fig. 7-6.2. The relative contributions of the systematicuncertainty, as given by ��ibXi

�2�(uK)2, and of therandom uncertainty, as given by ��isXi

�2�(uK)2, areshown and the pareto is ranked (left to right)

27


in terms of the combined standard uncertaintycontribution of each parameter to the result. Theanalysis shows that random errors in the volumet-ric flow rate measurement and systematic errorsin the pressure drop measurement are the largestcontributors to the overall uncertainty in loss coeffi-cient.



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Section 8Additional Uncertainty Considerations

8-1 CORRELATED SYSTEMATIC STANDARDUNCERTAINTIES

The expressions for the systematic standard un-certainty of the result in subsection 7-4 [eqs. (7-4.1)and (7-4.2)] assume that the systematic standarduncertainties of the measured parameters are allindependent of each other. There are many situa-tions where systematic errors for some of theparameters in a result are not independent. Exam-ples would include using the same apparatus tomeasure different parameters or calibrating differ-ent parameters against the same standard. In thesecases, some of the systematic errors are said tobe correlated and these nonindependent errorsmust be considered in the determination of thesystematic standard uncertainty of the result [9].

Consider an example where the result (R) isdetermined from three parameters (X1, X2, X3) thathave correlated systematic errors. The result iscalculated as

R p f(X1 , X2 , X3) (8-1.1)

and the absolute systematic standard uncertaintyof the result is given as

bR p ��1 bX1�2 + ��2 bX2

�2 + ��3 bX3�2

+2�1�2 bX1X2+ 2�1�3 bX1X3

(8-1.2)

+ 2�2�3 bX2X3�

1⁄2

The first three terms under the square root ineq. (8-1.2) are the same as those obtained by usingeq. (7-4.1), and the last three terms are those thataccount for the correlation among the systematicstandard uncertainties in X1, X2, and X3. Theterms bXiXk

are the estimates of the covariances ofthe systematic errors in Xi and Xk (see Nonmanda-tory Appendix B). These terms must be includedwhen systematic standard uncertainties for sepa-rate parameters, Xi and Xk, are from the same

28


source making them correlated, or thus their mea-surement errors are no longer independent. Theunits of the correlation terms (covariances),bXiXk

, are the product of the units of Xi and Xk.The covariance terms in eq. (8-1.2) must be

properly interpreted. Each bXiXk, term represents

the sum of the products of the portions of bXiand bXk

that arise from the same source and aretherefore perfectly correlated [10]. For instance, ifelemental systematic standard uncertainties 1 and2 for parameters 2 and 3 were from a commonsource, then bX2X3

, would be determined as

bX2X3p bX2

1

+ bX22

bX32

(8-1.3)

The example in eq. (8-1.2) can be expanded toany number of parameters by including the termfor each pair of parameters that has correlatedsystematic standard uncertainties. Therefore, an-other form of eq. (8-1.2) is

bR p �I

ip1(�i bi)2 + 2 �

I-1

ip1�

I

kpi+1�i�kbik (8-1.4)

where

bip systematic standard uncertainty inparameter i

bikp covariance between the systematicstandard uncertainties for the ith andkth parameters, calculated as follows:

bik p �L

lp1bil

bkl(8-1.5)

Ip number of parametersi and kp indexes indicating the ith and kth pa-

rameters�p sensitivity coefficients



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Table 8-1 Burst Pressures (Example 8-1-1)

Base Design, Improved Systematic StandardPb, Design, Pn, Uncertainty, bP,

106 Pa 106 Pa 106 Pa

Program 1Meter #1 40.0 . . . 0.2Meter #2 . . . 52.0 0.2

Program 2Meter #3 42.0 54.7 0.5

where

Lp number of common (correlated) errorsources

lp an index

Example 8-1-1: The use of back-to-back testsis an excellent method to reduce the systematicstandard uncertainty when comparing two or moredesigns. This method is a special case of correlatedsystematic standard uncertainties. Consider a bursttest for an improved container design. The im-provement in the design can be expressed as thefraction

R pPn

Pb

where

Pnp the burst pressure of the new designPbp the burst pressure of the original or base

design

Table 8-1 provides burst tests for two differentprograms. In the first test program, different pres-sure transducers were used in the tests on thetwo designs. There were no correlated systematicstandard uncertainties common between these twotransducers. In the second program, the samepressure transducer was used for both tests; there-fore, the systematic standard uncertainty was thesame and was correlated for the two test measure-ments.

Program 1 (no correlated systematic uncertain-ties):

Rp52.040.0

p 1.30

�bp−RPb

p −0.0325 (106 Pa)−1

29


�npRPn

p 0.0250 (106 Pa)−1

b2Rp [(−0.0325)(106 Pa)−1 (0.2)(106 Pa)]2

+ [(0.025)(106 Pa)−1 (0.2)(106 Pa)]2

bRp 0.0082

Program 2 (correlated systematic uncertainties):

Rp54.742.0

p 1.30

�bp −0.0310 (106 Pa)−1

�np 0.0238 (106 Pa)−1

b2Rp [(−0.0310)(106 Pa)−1 (0.5)(106 Pa)]2

+ [(0.0238)(106 Pa)−1 (0.5)(106 Pa)]2

+ 2(−0.0310)(106 Pa)−1 (0.0238)(106 Pa)−1

(0.5)(106 Pa)(0.5)(106 Pa)bRp 0.0036

This example demonstrates the strength of theback-to-back testing technique using the same in-strumentation. Even though the pressure trans-ducer in Program 2 had a systematic standarduncertainty of more than twice those of the trans-ducers in Program 1, the systematic standard un-certainty of the result for Program 2 was less thanhalf of that for Program 1.

Example 8-1-2 (adapted from [9]): Consider thepiping arrangement shown with the four flow-meters:

From conservation of mass, a balance checkwould yield

z p m4 − m1 − m2 − m3 p 0

If the errors in the flow rate measurements arepredominantly systematic, then for the balancecheck to be satisfied the absolute value of z mustbe less than or equal to the uncertainty in z

|z | ≤ 2bz



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Consider the case where the dominant systematicerrors are from the calibration standard and fromthe calibration curve fit. The calibration standardsystematic standard uncertainty for each flowmeteris ±1.5 kg/h for the three small meters and ±4.5kg/h for the large meter. The curve fit systematicstandard uncertainty for each meter is ±0.5 kg/h.Note that for this example the derivatives for eq.(8-1.2) are

�m1p �m2

p �m3p −1

and

�m4p 1

Case 1: Each Flowmeter Calibrated Against aDifferent Standard. In this case, all of the system-atic standard uncertainties are uncorrelated withthe systematic standard uncertainty for the threesmall flowmeters determined as

bmi(i p 1, 2, 3) p �(1.5)2 + (0.5)2�

1⁄2p 1.6 kg/h

and the systematic standard uncertainty for thelarge flowmeter calculated as

bm4p �(4.5)2 + (0.5)2�

1⁄2p 4.5 kg/h

Using eq. (7-4.1), the systematic standard uncer-tainty for z is

bz p ��m1bm1

�2 + ��m2bm2

�2 + ��m3bm3

�2 + ��m4bm4

�2�1 ⁄2

or

bz p ��bm1�2 + �bm2

�2 + �bm3�2 + �bm4

�2�1 ⁄2

p 5.3 kg/h

and the balance check will be satisfied if

|z | ≤ 2bz p 10.6 kg/h

Case 2: Flowmeters 1, 2, and 3 CalibratedAgainst the Same Standard and Flowmeter 4Calibrated Against a Different Standard. In thiscase, the systematic standard uncertainty for thethree small flowmeters that originates from thecalibration standard is correlated. The systematicstandard uncertainty from their curve fits is not,

30


because it is due to the scatter in the calibrationline. The final uncertainty is obtained as follows:

bm1p bm2

p bm3p 1.6 kg/h

bm4p 4.5 kg/h

and

bm1m2p bm1m3

p bm2m3p (1.5 kg/h)(1.5 kg/h)

Using eq. (8-1.4) for four parameters with threeof them having correlated systematic standard un-certainties, the systematic standard uncertainty forz becomes

bz p ��m1bm1

�2 + ��m2bm2

�2 + ��m3bm3

�2 + ��m4bm4

�2

+ 2�m1�m2

bm1m2+ 2�m1

�m3bm1m3

+ 2�m2�m3

bm2m3�

1⁄2

or

bz p ��bm1�2 + �bm2

�2 + �bm3�2 + �bm4

�2

+ 2bm1m2+ 2bm1m3

+ 2bm2m3�

1 ⁄2

bz p 6.4 kg/h


|z | ≤ 2bz p 12.9 kg/h

Note that in this case the signs for all thecorrelated terms are positive because all of thederivatives of z with respect to m1, m2, and m3are negative. If flowmeters 1, 2, and 3 are calibratedagainst the same standard, and flowmeter 4 iscalibrated against a different standard, the system-atic standard uncertainty for z is larger than if allthe meters had been calibrated against differentstandards (Case 1).

Case 3: Flowmeters 1, 2, 3, and 4 CalibratedAgainst the Same Standard. In this case, thesystematic standard uncertainties are

bm1p bm2

p bm3p 1.6 kg/h

and

bm4p 4.5 kg/h



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


with

bm1m2p bm1m3

p bm2m3p (1.5 kg/h)(1.5 kg/h)

and

bm1m4p bm2m4

p bm3m4p (1.5 kg/h) (4.5 kg/h)

Using eq. (8-1.4) for four parameters, all withcorrelated systematic standard uncertainties, thesystematic standard uncertainty for z is

bz p ��m1bm1

�2 + ��m2bm2

�2 + ��m3bm3

�2

+ ��m4bm4

�2 + 2�m1�m2

bm1m2

+ 2�m1�m3

bm1m3+ 2�m1

�m4bm1m4

+ 2�m2�m3

bm2m3+ 2�m2

�m4bm2m4

+ 2�m3�m4

bm3m4�

1⁄2

or

bz p ��bm1�2 + �bm2

�2 + �bm3�2 + �bm4

�2

+ 2bm1m2+ 2bm1m3

− 2bm1m4

+ 2bm2m3− 2bm2m4

− 2bm3m4�

1⁄2

bz p 1.0 kg/h


|z | ≤ 2bz p 2.0 kg/h

Note the signs for each of the correlated terms.For this case, calibrating all the flowmeters

against the same standard will yield the minimumsystematic standard uncertainty for z. It is interest-ing to note that the systematic standard uncertaintyin z is less than the smallest estimate of systematicstandard uncertainty for any of the flowmeters.

In general, correlated systematic standard uncer-tainties can either decrease, increase, or have noeffect on the systematic standard uncertainty ofthe result, depending on the form of the datareduction equation and on which parameters havecorrelated systematic errors.

31


8-2 NONSYMMETRIC SYSTEMATIC UNCERTAINTY

In some experiments, physical models can beused to essentially replace the asymmetric uncer-tainties with symmetric uncertainties in additionalexperimental variables. If this can be done thenit should be, but if not then the method of para.8-2.1 should be used. This paragraph presents amethod for determining nonsymmetric uncertaintyintervals in these cases [11].

8-2.1 Nonsymmetric Systematic UncertaintyInterval for a True Value

If the distribution of the systematic error associ-ated with a measured variable is symmetricallydistributed but not centered at zero, then theoverall uncertainty interval for the unknown truevalue will not be centered on the measured valueof the variable. In this case, the following procedureshould be employed for constructing a nonsymmet-ric uncertainty interval for the unknown true valueof the quantity being measured (see Fig. 8-2.1):

(a) Specify an interval (X − B−, X + B+) relative tothe measured value X within which one may expectthe true value to fall with 95% confidence, in theabsence of random errors. This interval accounts forsystematic errors only.

(b) Define the offset, q, as the difference betweenthe center of the interval specified in (a) and themeasured value. Thus

q p(X + B+) + (X − B−)

2− X p

B+ − B−

2

(c) Define the quantity B as follows:

B p(X + B+) − (X − B−)

2p

B+ + B−

2

Thus B is equal to one-half the length of theinterval specified in (a).

(d) Calculate bX, the systematic standard uncer-tainty for the measurement, as

bX pB2



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Fig. 8-2.1 Schematic Relation Between Parameters Characterizing Nonsymmetric Uncertainty

This is based on the assumption that a Gaussiandistribution is an appropriate model for the system-atic error. (See Nonmandatory Appendix B forother distributional models.)

(e) Calculate uX, the combined standard uncer-tainty for the measurement, using the standard for-mula uX p �b 2

X+ s2

X.

(f) Calculate U95, the expanded uncertainty forthe measurement, using

U95 p 2uX p 2�b 2X + s 2

X (8-2.1)

This calculation is based on the assumption thatthe degrees of freedom for the combined standarduncertainty are large. (For small degrees of freedomsee Nonmandatory Appendix B.)

(g) Calculate an approximate 95% confidence in-terval for the true value using

�X + q� ± U95 (8-2.2)

(h) Express the final result as an asymmetric 95%confidence interval for the true value with the lowerlimit given by

X lower limit p X + q − U95 p X − U− (8-2.3)

and the upper limit given by

Xupper limit p X + q + U95 p X + U+ (8-2.4)

where U− p U95 − q and U+ p U95 + q.

32


Example 8-2.1: Suppose a thermocouple is beingused to measure the temperature of a gas stream,but the user of the thermocouple believes theremay be a tendency for the thermocouple to providea temperature reading that is lower than the actualgas temperature due to a radiative heat transfermechanism. The user does not have enough infor-mation to properly correct the thermocouple read-ing for these effects, but wishes to account forthem in an uncertainty analysis. From a sampleof more than 30 readings using the thermocouple,the user finds that X p 534.7°C and sX p 2.4°C.If the user believes that the true gas temperaturemay be as much as 10°C higher than X due toradiation effects, then a nonsymmetric confidenceinterval accounting for this nonsymmetric system-atic uncertainty may be computed as follows:

(a) Specify an interval (corresponding to 95% con-fidence) for the systematic error in question. In thiscase, the user of the thermocouple believes that thetrue gas temperature falls between the average mea-sured with the thermocouple, X p 534.7°C, and avalue that is 10°C higher than X, i.e., 544.7°C. SoB− p 0°C and B+ p 10°C.

(b) Determine q, the difference between the centerof the interval specified in (a) and the value mea-sured with the thermocouple. In this case

q p (544.7°C + 534.7°C)/2 − 534.7°C p 5°C



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


(c) Calculate the quantity B as

B pB+ + B−

2p

10°C + 0°C2

p 5°C

(d) Calculate bX, the systematic standard uncer-tainty for the measurement, as

bX pB2

p 2.5°C

This is based on the assumption that a Gaussiandistribution is an appropriate model for the system-atic error.

(e) Calculate uX, the combined standard uncer-tainty for the measurement, using the standardformula

uX p �b 2X + s2

X p �(2.5°C)2 + (2.4°C)2 p 3.45°C

(f) Calculate U95, the expanded uncertainty forthe measurement, using

U95 p 2uX p 6.9°C

This calculation is based on the assumption thatthe degrees of freedom for the combined standarduncertainty are large. (For small degrees of freedomsee Nonmandatory Appendix B).

(g) Calculate an approximate 95% confidence in-terval for the true value using

�X + q� ± U95

In this case, this 95% confidence interval is given by

[534.7°C + 5°C] ± 6.9°C

(h) Calculate

U− p U95 − q p 6.9°C − 5°C p 1.9°C

and

U+ p U95 + q p 6.9°C + 5°C p 11.9°C

The final result may be expressed as an asymmet-ric 95% confidence interval for the true value usingthe lower limit given by

Xlower limit p X − U− p 534.7°C − 1.9°C p 532.8°C

33


and the upper limit given by

Xupper limit p X + U+ p 534.7°C + 11.9°C p 546.6°C

8-2.2 Nonsymmetric Systematic UncertaintyInterval for a Derived Result

A nonsymmetric systematic uncertainty in ameasured variable may also result in a nonsymmet-ric uncertainty interval for a derived result. Thefollowing procedure may be employed for propa-gating the nonsymmetric uncertainties in a set ofmeasured variables to a derived result (see Fig.8-2.2):

(a) Determine Xi, uXi, and qi for each average Xi

that contributes to the determination of the derivedresult, r�X1, X2, . . ., Xn�.

(b) Determine the offset, qr, which is defined as

qr p r�X1 + q1, X2 + q2, . . ., Xn + qn�− r�X1, X2, . . ., Xn�.

(c) Determine the sensitivity coefficient, �i, foreach average X; that contributes to the derived resultfollowing standard procedure. If a sensitivity coeffi-cient depends on the values of any averages, i.e.,�i p �i�X1, X2, . . ., Xn�, then it should be evaluatedat the point �X1 + q1, X2 + q2, . . ., Xn + qn�.

(d) Calculate ur, the combined standard uncer-tainty for the derived result, using the standardformula:

(8-2.5)ur p ��1uX1�2 + ��2uX2

�2 + . . . + ��nuXn�2�

(e) Calculate U95,r the expanded uncertainty forthe derived result at a 95% confidence level, as

U95,r p 2ur

(This is based on the assumption that the degreesof freedom are large. For small degrees of freedom,see Nonmandatory Appendix B.)

(f) Calculate an approximate 95% confidence in-terval for the derived result using

r�X1 + q1 , X2 + q2 , . . . , Xn + qn� ± U95,r (8-2.6)



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Fig. 8-2.2 Relation Between Parameters Characterizing Nonsymmetric Uncertainty

(g) Express this confidence interval as an asym-metric 95% confidence interval for the derived resultas follows:

r�X1 , X2 , . . . , Xn� ± �U95,r ± qr� (8-2.7)

where the lower limit on this interval is given by

r lower limit p r(X1 , X2, . . . , Xn)

− (U95,r − qr)

p r − Ur− (8-2.8)

and the upper limit on this interval is given by

rupper limit p r(X1 , X2, . . . , Xn)

+ (U95,r + qr)

p r + Ur+ (8-2.9)

with Ur− p U95,r − qr and U+

r p U95,r + qr.Example 8-2.2: Suppose the user of the thermo-

couple in the example in para. 8-2.1 wishes to usethis gas temperature to estimate the speed of soundfor the gas using the following relation:

c p [kRT]1⁄2

where k, the ratio of specific heats, and R, the gasconstant for the gas, are taken to be constant with

34


negligible uncertainty and T is the measured valueof the absolute temperature. In this case, T p(X + 273.2)K, where X is the average value of thetemperature of the gas using the thermocouple.The uncertainty interval for c may be calculatedas follows:

(a) Determine T, uT, and qT for the measured vari-able T. In this case, T p 807.7K, uT p 3.45K, andqT p 5K.

(b) Determine the offset, qc, as follows:

qc p c(T + qT) − c(T) p [kR(812.7K)]1 ⁄2

− [kR(807.7K)]1 ⁄2 p [kR]

1⁄2 (0.0878K)1 ⁄2

(c) Determine the sensitivity coefficient, �T, forthe measured variable T. In this case, �T p(1/2)[kR/T]

1⁄2. Since this sensitivity coefficient de-pends on T, it should be evaluated at T + qT p812.7K, so that here

�T p (1 ⁄2)[kR/(812.7K)]1⁄2 p [kR]

1⁄2 (0.0175K)−1⁄2

(d) Estimate the combined standard uncertaintyfor the derived result, uc. In this case,

uc p [{(kR)1 ⁄2 (0.0175K−1⁄2 (3.45K)}2]

1 ⁄2

p [kR]1⁄2 (0.0604K)

1⁄2



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


(e) Calculate U95,c the expanded uncertainty forthe derived result c, at a 95% confidence level, as

U95,c p 2uc p 2[kR]1 ⁄2 (0.0604K)

1⁄2 p [kR]1⁄2 (0.1208K)

1 ⁄2

(This is based on the assumption that the degreesof freedom are large. For small degrees of freedom,see Nonmandatory Appendix B.)

(f) Compute a 95% confidence interval for thederived result using c(T + qT) ± U95,c. In this case,this 95% confidence interval is given by

[kR]1⁄2 [812.7K]

1⁄2 ± [kR]1 ⁄2 (0.1208K

1 ⁄2)

(g) Express the final result as an asymmetric 95%confidence interval using

c(T) ± (U95,c ± qc)

In this case, this 95% confidence interval is given by

[kR]1⁄2 [807.7K]

1⁄2 ± {[kR]1⁄2 (0.1208K

1⁄2)

± [kR]1⁄2(0.0878K

1 ⁄2)}

whose lower limit is equal to

clower limit p [kR]1 ⁄2 [28.42K

1 ⁄2] − [kR]1⁄2 (0.033K

1⁄2)

and whose upper limit is equal to

cupper limit p [kR]1 ⁄2 [28.42K

1 ⁄2] + [kR]1⁄2 (0.2086K

1 ⁄2)

In this example, the uncertainty interval for thespeed of sound of the gas extends from 0.12%below to 0.73% above the value for the speed ofsound assessed using the measured value of thetemperature.

8-3 FOSSILIZATION OF CALIBRATIONS

Definition of the Measurement Process is a pre-requisite for determining measurement uncertaintyestimates. For example, different Defined Measure-ment Processes for the same test will result indifferent estimates of measurement uncertainty.Elemental errors are classified as random if theyadd scatter to a result. If they do not, they aresystematic. Final classification is dependent on theDefined Measurement Process.

Calibration errors are frequently reclassified asa result of the Defined Measurement Process. As anexample, for test measurements involving multiple

35


calibrations (i.e., calibrations performed prior toeach of the multiple tests or periodically through-out a test program), the calibration process randomerror will introduce scatter in the test data setand therefore should remain classified as a randomerror source or random standard uncertainty inthe uncertainty analysis. However, for test mea-surements involving a single calibration (i.e., cali-brations performed only once during a test pro-gram, and all data samples processed using thesame calibration constants), the calibration processrandom error does not have an opportunity tointroduce scatter in the test data set and thereforeshould be reclassified (fossilized) as systematicerror during the final uncertainty analysis. In thiscase, the calibration random standard uncertaintyshould be treated as another elemental systematicstandard uncertainty and combined with the othercalibration systematic standard uncertainties to ob-tain the total systematic standard uncertainty forthe calibration process as:

b*c p �(b1)2 + (b2)2 + . . . (sc)2 (8-3.1)

whereb*

cp total systematic standard uncertaintyof the single calibration process

b1, b2, . . .p elemental systematic standard uncer-tainty components of the calibrationprocess

Ncp number of calibration points withina single calibration

scp standard deviation of the calibrationprocess; an estimated standard devi-ation of the random error compo-nents of the calibration process

scpsc��Nc

; the fossilized elemental sys-tematic standard uncertainty due tothe random error of the calibrationprocess

A more general form of eq. (8-3.1) for a singlecalibration is

b*c p �(bc)2 + (sc)2�

1⁄2(8-3.2)

where

bcp calibration process initial systematic stan-dard uncertainty before reclassifying (fos-silizing) the calibration random standarduncertainty into systematic standard un-certainty



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


To illustrate the fossilization process, considera master flowmeter installed in line with a testflowmeter for use in establishing a calibrationcorrection for the test meter. As a result of thecalibration process, the systematic standard uncer-tainty of the test meter is replaced by that ofthe master meter. The calibration process randomstandard uncertainty is a function of the randomstandard uncertainties in both the master and testmeters. When a data set of interest from the testmeter involves multiple calibrations, the calibrationprocess random standard uncertainty will causescatter in the individual test meter data samplesand thus should remain classified as calibrationrandom standard uncertainty in the uncertaintyanalysis. However, when a data set from the testmeter involves only a single calibration, the calibra-tion process random standard uncertainty is com-mon to all data samples and thus manifests itselfin the data set as a systematic standard uncertainty(becomes fossilized). In this case, the random andsystematic standard uncertainties of the calibrationprocess should be combined and carried forwardas a systematic standard uncertainty.

Fossilization of calibration random standard un-certainty can occur at any or all levels of calibrationhierarchy, from a national laboratory to the testapplication, depending on the defined calibration,or measurement processes, or both. Sometimes,multiple calibrations are performed but the resultsare averaged into a single set of calibration con-stants for use in processing all data samples (e.g.,pre- and post-calibrations). In this case, a portionof the random standard uncertainty still becomesfossilized into systematic standard uncertainty. Themagnitude has been reduced by having averagedmultiple calibrations. The term sc in eq. (8-3.2)should be reduced by dividing by �Nrc

. Thus amore general form of eq. (8-3.2) would be

bc* p (bc) + �sc/�Nrc�21 ⁄2

(8-3.3)

where

Nrcp number of repeated independent (single)calibrations averaged and used in ob-taining a single set of calibration constantscommon to all samples within the testdata set

36


8-4 SPATIAL VARIATION [12]

Measurement requirements for a performancetest are often such that an average measurementof individual parameters is required. Most instru-mentation, however, yields a point measurement ofa parameter rather than an average measurement.While this point characteristic may be useful forother purposes, it raises a problem in determiningperformance level. In many instances, the quantitymeasured varies in space, making the point mea-surement inadequate. Thus, it often is necessaryto install several measurement sensors at differentspatial locations to account for spatial variationsof the parameter being measured. Spatial variationeffects are considered errors of method (see para.5-2.5).

A simple illustration of the impact of spatialvariation is the measurement of average velocityin fully developed flow of an incompressible fluidin a pipe. At low Reynolds numbers, the flow islaminar and has a parabolic velocity profile. Onemeasurement will not give the true average valuedirectly. For example, the velocity at the centerof the pipe is twice the average velocity. This is incontrast to the situation at high Reynolds numbers,where the flow is turbulent. At higher Reynoldsnumbers, the profile approaches uniformity andany measurement will yield a reasonable estimateof the average velocity. Usually, test conditionsvary between these two extremes and it is notpossible to correct the readings in a simple manner.Circumferential variations may also be present.Therefore, other approaches are chosen, such asinstalling multiple sensors and averaging theoutputs.

An example of an averaged output is an averag-ing pressure probe often used to measure averagevelocity at a cross section of pipe. The uncertaintyof this mean velocity would be calculated byconsidering it to be a determined result. An addi-tional systematic standard uncertainty may needto be assigned to the mean result to account forthe possible difference between the determinedmean and the true mean.

8-5 ANALYSIS OF REDUNDANT MEANS

When redundant instrumentation or calculationmethods are available, the individual results andtheir uncertainties should be compared with eachother and with the pretest uncertainty analysis.



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Fig. 8-5.1 Three Posttest Cases

When comparing redundant means (X1 andX2) and their uncertainty intervals, the three casesillustrated in Fig. 8-5.1 need to be considered.

Case 1: A problem clearly exists when there isno overlap between uncertainty intervals. Eitheruncertainty intervals have been grossly underesti-mated, or the true value is not constant. Investiga-tion to identify bad readings, overlooked or under-estimated systematic uncertainty, etc., is necessaryto resolve this discrepancy.

Case 2: When the uncertainty intervals com-pletely overlap, as in Case 2, one can be reasonablyconfident that there has been a proper accountingof all major uncertainty components. The smalleruncertainty interval X2 ± U2 is wholly containedin the larger interval X1 ± U1 Since the individualmeasurements are valid, the weighting methoddescribed in para. A-2 may be used to obtain abetter estimate of the true value than either ofthe individual measurements.

Case 3: Case 3, where a partial overlap of theuncertainty intervals exists, is the most difficultto analyze. For both measurements and both uncer-tainty intervals to be correct, the true value mustlie in the region where the uncertainty intervalsoverlap. Consequently, the larger the overlap, themore confidence we have in the validity of the

37


measurements and the estimate of the uncertaintyintervals. As the difference between the two mea-surements increases, the overlap region shrinks.Standard statistical hypothesis testing may be usedto evaluate the significance of the difference ob-served. The process is outlined in the followingparagraphs.

Let s1 and b1 denote, respectively, the randomand systematic standard uncertainties associatedwith X1. Likewise, let s2 and b2 denote, respectively,the random and systematic standard uncertaintiesassociated with X2. Assuming that the degrees offreedom associated with the systematic and ran-dom uncertainty components are large, and thatthere are no correlated errors, the Z-statistic fortesting the null hypothesis that the two measure-ments X1 and X2 have an expected difference ofzero (i.e., the two measurements are consistentwith one another) is given by

Z pX1 − X2

�s21 + s2

2 + b21 + b2

2�1⁄2

If |Z| > 2, then the data indicate, at the 95%confidence level, that the two measurements areconsistent with one another. Otherwise, there is



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


no compelling evidence to declare the two mea-surements to be inconsistent with one another.

Caution: Lack of evidence for demonstrating inconsistencydoes not “prove” consistency.

8-6 REGRESSION UNCERTAINTY

8-6.1 Linear Regression Analysis

Curve fitting often is used in the calibrationprocess, in the data reduction program, and inthe representation of the final test results. Least-squares regression analysis is the most popularmeans of curve fitting. In many cases, the antici-pated representation of the data is a straight line,or a simple (first-order) linear regression. In othercases, the data to be curve-fit often can be rectified,or transformed, into linear coordinates [9, 13, 14].

Subsection 8.6 will cover only straight-line re-gressions to estimate the relationship for Y versusX. For higher-order linear regressions and otherregression methodologies, see Refs. [9, 15, 16, 17,18]. For regression uncertainty when X and Y arefunctions of other variables, see Refs. [9, 17].

The random standard uncertainty for the curve-fit will be determined using standard least squaresanalysis [9, 14, 16], where the assumption is madethat there is no random standard uncertainty inthe X values and the random standard uncertaintyin the Y values is constant over the range of thecurve-fit. In this section, only a special case isconsidered for the systematic standard uncertainty.This special case is where the systematic standarduncertainty for the Y values and/or the X valuesis a constant (i.e., percent of full scale) and thereare no correlated elemental systematic standarduncertainties between the X and Y values. A moregeneral approach to regression uncertainty is pre-sented in [9] where the methodology applies forvariable random standard uncertainties in X andY, variable systematic standard uncertainties in Xand Y, and correlated systematic standard uncer-tainties between X and Y.

8-6.2 Least-Squares

For a straight-line, or a simple linear regression,the curve-fit expression is

Y p mX + c (8-6.1)

38


where for N data pairs (Xj, Yj), the slope m isdetermined from

m p

N �N

jp1XjYj − �

N

jp1Xj �

N

jp1Yj

N �N

jp1(X2

j ) − � �N

jp1Xj�

2(8-6.2)

and the intercept c is determined from

c p

�N

jp1(X2

j ) �N

jp1Yj − �

N

jp1Xj �

N

jp1(XjYj)

N �N

jp1(X2

j ) − � �N

jp1Xj�

2(8-6.3)

The least-squares process essentially providesan average for the data so that the regressionexpression in eq. (8-6.1) represents the relationshipbetween the mean value of Y and X. This mean,Y, is not the average of the Yj data but the meanY response from the curve-fit for a given X. Oncethe slope and intercept are calculated from eqs.(8-6.2) and (8-6.3), these constants can be substi-tuted into eq. (8-6.1) along with several values ofX and the resulting straight line can be plottedover the (Xj, Yj) data. Since the Y vs. X curve isa mean value for the data set, the curve shouldbe a good representation of the data if the simplelinear fit is appropriate.

8-6.3 Random Standard Uncertainty for YDetermined From Regression Equation

The statistic that defines the standard deviationfor a straight-line curve-fit is the standard errorof estimate

SEE p �N

jp1(Yj − mXj − c)2

N − 2

1 ⁄2

(8-6.4)

For a given value of X, the random standarduncertainty associated with the Y obtained fromthe curve-fit [eq. (8-6.1)] is

sY p SEE 1N

+(X − X)2

�N

jp1(Xj − X)2

1 ⁄2

(8-6.5)



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


where

X p1N �

N

jp1Xj (8-6.6)

If there is no random standard uncertainty inthe Xj data or the new X values used in theregression equation, the random standard uncer-tainty sY obtained from eq. (8-6.5) is combined withthe systematic standard uncertainty (discussed inpara. 8-6.4 using eq. (7-5.1) to obtain the totaluncertainty for the Y value from the curve-fit. Forrandom standard uncertainty in the Xj or X values,the general approach in [9] should be used.

8-6.4 Systematic Standard Uncertainty for YDetermined From Regression Equation

There can be systematic standard uncertainty,bYj

and bXjrespectively, in the Yj and Xj data. There

also can be systematic standard uncertainty in theX value used in the curve-fit to find a Y value.This curve-fit X will be called Xnew to distinguishit from the Xj data points, and the systematicstandard uncertainty for Xnew is bXnew

. It is verylikely that most, and probably all, of the elementalsystematic standard uncertainties for each of theYj data points are from the same sources, and are,therefore, correlated. The same is true for the Xjdata points. There is also the possibility that theXnew values will have systematic standard uncer-tainties from the same sources as the Xj datacausing these uncertainties to be correlated.

In subsection 8-6, only constant systematic stan-dard uncertainties for (Xj, Yj) and Xnew are consid-ered. All of the bYj uncertainties are assumed tobe completely correlated with each other, and allof the bXj

uncertainties are assumed to be com-pletely correlated with each other. The assumptionis made that there are no common uncertaintysources between Yj and Xj (no correlation betweenthe bYj

and bXjsystematic standard uncertainties).

Cases are considered where Xnew has systematicstandard uncertainty which is correlated with thatin Xj and where Xnew has systematic standarduncertainty which is not correlated with that in Xj.

8-6.4.1 Systematic Standard Uncertainty in Yj

Data. If each of the Yj data points has the samesystematic standard uncertainty, bY1

, then the re-sulting elemental systematic standard uncertaintyfor the mean Y from the curve-fit is [9, 14, 17]

39


bY1p bY1

(8-6.7)

8-6.4.2 Systematic Standard Uncertainty in Xj

Data With No Systematic Standard Uncertainty inXnew. If each of the Xj data points has the samesystematic standard uncertainty, bX, and Xnew hasno systematic standard uncertainty, then the re-sulting elemental systematic standard uncertaintyfor the mean Y from the curve-fit is determinedas [9, 17]

bY2p mbx (8-6.8)

This case would occur when the regression equa-tion from a set of test data is used later in adesign or analysis process where Xnew might betaken as a value that has no uncertainty.


Data With Correlated Systematic Standard Uncer-tainty in Xnew. If each of the Xj data points has thesame systematic standard uncertainty, bX, and Xnewhas the same systematic standard uncertainty (fromthe same sources), then the resulting elementalsystematic standard uncertainty for the mean Yfrom the curve-fit is zero [9]. This case wouldoccur if the same instruments are used to measureXnew as were used to measure Xj. Since all of thesystematic standard uncertainties for Xj and Xneware correlated, the systematic standard errors areall the same. The effect on the curve-fit is to shiftit to the right or left depending on the sign ofthe errors (the signs and magnitudes of the errorsare unknown). This shift has no effect on the valueof Y obtained from the curve since the shift inXnew is the same as the shift in Xj.


Data With Uncorrelated Systematic Standard Uncer-tainty in Xnew. If each of the Xi data points hasthe same systematic standard uncertainty, bX, butXnew has a different (no common systematic errorsources) systematic standard uncertainty, bXnew

,then the resulting elemental systematic standarduncertainty for the mean Y from the curve-fit is [9]

bY3p �(mbX)2 + (mbXnew)2�

1 ⁄2(8-6.9)

This case would occur if different instrumentswere used to measure the Xj values and Xnew.



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Table 8-6.4.5 Systematic Standard UncertaintyComponents for Y Determined from Regression

Equation

Systematic standard uncertaintyin Yj data bY

1p bY

1

Systematic standard uncertaintyin Xj data with no systematicstandard uncertainty in Xnew bY

2p mbX

Systematic standard uncertaintyin Xj data with correlatedsystematic standarduncertainty in Xnew 0

Systematic standard uncertaintyin Xj data with uncorrelatedsystematic standarduncertainty in Xnew bY

3p ��mbX�2 + �mbXnew�2�

1⁄2

8-6.4.5 Systematic Standard Uncertainty for Y.The systematic standard uncertainty for the meanY from the curve-fit will be the appropriate root-sum-square of the bYi

elemental systematic stan-dard uncertainties defined above and summarizedin Table 8-6.4.5.

For systematic standard uncertainty in Yj onlyor systematic standard uncertainty in Yj with corre-lated systematic standard uncertainty between Xjand Xnew

bY p bY1(8-6.10)

For systematic standard uncertainty in the Yjdata and the Xj data and no systematic standarduncertainty in Xnew

bY p �b2Y1

+ b2Y2

�1⁄2

(8-6.11)

For systematic standard uncertainty in the Yjdata and the Xj data and uncorrelated systematicstandard uncertainty in Xnew, the systematic stan-dard uncertainty for the curve-fit value of Y is

40


bY p �b2Y1

+ b2Y3

�1⁄2

(8-6.12)

For no systematic standard uncertainty in theYj data, systematic standard uncertainty in theXj data, and no systematic standard uncertaintyin Xnew

bY p bY2(8-6.13)

For no systematic standard uncertainty in theYj data, systematic standard uncertainty in the Xjdata, and uncorrelated systematic standard uncer-tainty in Xnew, the systematic standard uncertaintyfor the curve-fit value of Y is

bY p bY3(8-6.14)

For no systematic standard uncertainty in theYj data and correlated systematic standard uncer-tainty between Xj and Xnew

bY p 0 (8-6.15)

8-6.5 Uncertainty for Y From RegressionEquation

The total uncertainty in the Y obtained fromthe simple linear regression expression, eq. (8-6.1),is given by eq. (7-5.1) for the case where thedegrees of freedom for Y are sufficiently large sothat t ≈ 2

UY p 2�b2Y + s

2Y�

1⁄2(8-6.16)

Note that the degrees of freedom for Y is basedon the degrees of freedom for sY, which is N −2, and the degrees of freedom for bY (see Nonman-datory Appendix B). The use of the factor t ≈ 2will be appropriate in most cases. The uncertaintyband UY in eq. (8-6.16) will vary with X (i.e., Xnew)because of the expression for sY from eq. (8-6.5).As noted earlier, the uncertainty expression in eq.(8-6.15) only applies if there is no random standarduncertainty in X and if the systematic standarduncertainties are percent of full-scale values.



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Section 9Step-by-Step Calculation Procedure

9-1 GENERAL CONSIDERATIONS

It is recommended that an uncertainty analysis,following the methods of Sections 4 through 8 ofthis Supplement, be conducted before and aftereach test, according to the procedure that follows.The pretest analysis (see subsection 4-4) is usedto determine if the test result can be measured withsufficient accuracy, i.e., the predicted uncertaintyshould be smaller than the required uncertainty.It may also be used to compare alternative instru-mentation systems and test designs and to deter-mine corrective action if the predicted uncertaintyis unacceptably large. Furthermore, it may be usedto evaluate the need for calibration. The posttestanalysis (see subsection 4-4) validates the pretestanalysis, provides data for validity checks, andprovides a statistical basis for comparing test re-sults.

9-2 CALCULATION PROCEDURE

(a) Define Measurement Process (see section 5).(1) Review test objectives and test duration.(2) List all independent measurement parame-

ters and their nominal levels.(3) List all calibrations and instrument setups

that will affect each parameter. Be sure to check foruncertainties in measurement system componentsthat affect two or more measurements simultane-ously (correlated uncertainties).

(4) Define the functional relationship betweenthe independent measurement parameters and thetest result.

(b) List Elemental Error Sources (see subsection5-3).

(1) Make a complete and exhaustive list of allpossible test uncertainty sources for all parameters.

(c) Calculate the Systematic Uncertainty and Ran-dom Uncertainty (Standard Deviation of the Mean)for Each Parameter (see subsections 6-1 and 6-2).

(d) Propagate the Systematic and Random Stan-dard Deviations (see subsections 7-1 through 7-4).

41


(1) The systematic uncertainty and random un-certainty (sample standard deviations of the means)of the independent parameters are propagated sepa-rately all the way to the final result.

(2) Propagation of the standard deviations ofthe means is done, according to the functional rela-tionship defined in step (a)(4), by using the Taylorseries method (see section 7). This requires a calcula-tion of the sensitivity factors, either by differentia-tion or by numerical analysis.

(e) Calculate Uncertainty (see subsection 7-5).(1) Combine the systematic and random uncer-

tainties to obtain the total uncertainty.(f) Report.

(1) The uncertainty analysis for each calculatedresult should be reported on two tables. The first isa detailed report that displays all the informationused in the calculation of the nominal value anduncertainty of the result. The second is a table thatsummarizes the uncertainty information at the re-sult level. For most uncertainty analyses, all mea-sured parameters will have symmetric systematicuncertainties and large degrees of freedom. Forsome analyses, one or more of the systematic uncer-tainties may be nonsymmetric (see subsection 8-2),and for other analyses, the degrees of freedom maybe small for some of the uncertainties (see Nonman-datory Appendix B). The detailed report tableshould include, as a minimum, the following infor-mation for each parameter used in the calculationof the result:

(a) symbol used in the calculations(b) description(c) units(d) nominal value (average of measure-

ments), X(e) systematic standard uncertainty, bi(f) sample random standard uncertainty,

standard deviation of the mean, sx,i(g) sensitivity, ��

(h) systematic standard uncertainty contri-bution to the combined uncertainty of the result,(�ibi)2



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Table 9-2-1 Table of Data




Nominal Uncertainty, Uncertainty, Sensitivity, Contribution, Contribution,Symbol Description Units Value bX

iSX

i�i �ibX

i

2 �iSXi

2

C Discharge coefficient . . . 0.984 0.00375 0.0 140 0.276 0.0d Throat diameter in. 3.999 0.0005 0.0 86.2 0.00186 0.0D Inlet diameter in. 6.001 0.001 0.0 −11.4 0.00013 0.0� Water density at lbm/ft3 62.4 0.002 0.002 1.11 0.0000049 0.0000049

60°Fh Differential pressure in. H2O 100 0.15 0.4 0.6919 0.0108 0.0766

head acrossventuri (at 68°F)

Table 9-2-2 Summary of Data

Calculated Result

Absolute Absolute AbsoluteSystematic Random Combined Absolute

Calculated Standard Standard Standard ExpandedResult, Uncertainty, Uncertainty, Uncertainty, Uncertainty,

Symbol Description Units R bR sR uR UR

m Mass flow lbm/s 138.4 0.276 0.0766 0.286 0.572rate

(i) random standard uncertainty contribu-tion to the combined uncertainty of the result,(�isX)2

42


The summary report table should display theinformation associated with the result as detailedin Tables 9-2-1 and 9-2-2 which are based onExample 10-2.



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Section 10Examples

10-1 FLOW MEASUREMENT USING PITOT TUBES

10-1.1 Define the Measurement Process

The flow rate of an incompressible fluid ina pipe may be determined by multiplying theintegrated-average velocity of the fluid by thecross-sectional flow area of the pipe. One techniquefor measuring the integrated-average velocity ofthe fluid is to traverse the cross-sectional flowarea with a Pitot tube. Measurements at eachtraverse point can be used to determine local fluidvelocity. Traverse points are typically specified atthe centroid of equal areas so that the integrated-average velocity may be estimated as the averageof the measured values for all traverse points.This avoids the need to develop weighting factorsfor each sample area.

For this example, the velocity is measured at40 unique traverse points (10 traverse positionsalong 4 equally spaced radii) corresponding to thecentroid of equal areas as shown in Fig. 10-1.1.A total of 60 measurements are taken in successionat each traverse point once the Pitot tube is posi-tioned. The point velocity values at individualtraverse points are treated as measurements (Sec-tion 6); the average velocities calculated from thesepoint velocities are treated as results (Section 7).

Several simplifying assumptions are made forthis example:

(a) the pipe diameter is large compared to thePitot tube diameter such that blockage effects andwall interference effects can be neglected;

(b) the velocity pressure developed across the pi-tot tube is measured by a differential pressure trans-mitter;

(c) the output of the differential pressure trans-mitter is measured and recorded by a computerizeddata acquisition system (DAS);

(d) the DAS computes velocity for each measure-ment by taking the square root of the output of thedifferential pressure transmitter, making the appro-priate corrections for fluid density, and making the

43


appropriate calibration corrections and unit conver-sions;

(e) the DAS automatically takes 60 readings ateach traverse point and computes and records aver-age values and standard deviations for the data col-lected at each traverse point;

(f) the Pitot tube, differential pressure transmit-ter, and DAS are calibrated together as a system; and

(g) the flow rate and the velocity profile remainconstant for the duration of the test.

10-1.2 Data Summary

The computerized data acquisition system isused to compute average values from the 60 mea-surements at each traverse point using eq. (4-3.1).The resulting average values at each traverse pointXij are summarized in Table 10-1.2.

10-1.3 Velocity Results

The DAS is also programmed to output thesample standard deviation of the 60 measurementsat each traverse point based on eq. (4-3.2). Thesample standard deviation at each point is summa-rized in Table 10-1.3-1.

The traverse points are located at the centroidof equal areas so that the integrated-average veloc-ity in the pipe is approximated by the average ofthe velocities determined at the traverse points.First, the average velocity along each radius, Vi,is approximated as

Vi (ft/sec) ≈ �10

jp1(1⁄10) Xif (ft/sec) (10-1.1)

Then, the average velocity in the pipe V isapproximated as

V (ft/sec) ≈ �4

ip1(1⁄4) Vi (ft/sec) (10-1.2)



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Fig. 10-1.1 Traverse Points (Example 10-1)

Table 10-1.2 Average Values (Example 10-1)

Radius 1, i � 1 Radius 2, i � 2 Radius 3, i � 3 Radius 4, i � 4Traverse Point, j Xij (ft/sec) Xij (ft/sec) Xij (ft/sec) Xij (ft/sec)

j p 1 5.31 5.27 5.21 5.00j p 2 5.46 5.53 5.25 5.16j p 3 5.55 5.61 5.37 5.31j p 4 5.63 5.68 5.47 5.42j p 5 5.65 5.74 5.58 5.50j p 6 5.69 5.77 5.62 5.55j p 7 5.73 5.79 5.65 5.63j p 8 5.74 5.76 5.65 5.65j p 9 5.76 5.75 5.70 5.65j p 10 5.72 5.80 5.70 5.67

44




--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Table 10-1.3-1 Standard Deviations (Example 10-1)

Radius 1, i � 1 Radius 2, i � 2 Radius 3, i � 3 Radius 4, i � 4

SXij

SXij

SXij

SXij

SXij

SXij

SXij

SXij

Traverse Point, j (ft/sec) (ft/sec) (ft/sec) (ft/sec) (ft/sec) (ft/sec) (ft/sec) (ft/sec)

1 1.21 0.156 1.31 0.169 1.61 0.208 1.41 0.1822 1.06 0.157 1.61 0.208 1.78 0.230 1.65 0.2133 1.03 0.133 1.36 0.176 1.89 0.244 1.26 0.1634 1.21 0.156 1.31 0.169 1.84 0.238 1.80 0.2325 1.29 0.167 1.06 0.137 1.65 0.213 2.04 0.2636 1.09 0.141 1.26 0.163 1.09 0.141 1.74 0.2257 0.81 0.105 1.03 0.133 1.43 0.185 1.61 0.2088 1.00 0.129 0.93 0.120 1.18 0.152 2.14 0.2769 1.15 0.148 1.34 0.173 1.36 0.176 1.43 0.18510 0.81 0.105 1.45 0.187 1.00 0.129 1.54 0.199

Table 10-1.3-2 Summary of Average VelocityCalculation (Example 10-1)

Parameter Value (ft/sec)

V1 5.62V2 5.67V3 5.52V4 5.45V 5.57

The subscripts i and j are used in the previousequations to designate radius and traverse posi-tions, respectively.

The results of these calculations are summarizedin Table 10-1.3-2.

10-1.4 List Elemental Uncertainty Sources

The sources of uncertainty which are consideredrandom in this simplified example are those caus-ing variation in the 60 repeated measurements ofvelocity at each traverse point. The sources ofuncertainty which are considered systematic inthis simplified example are the uncertainty of thecalibration of the instruments used to measureand record velocity at each traverse point and theuncertainty of the integrated-average velocity dueto spatial variation.

10-1.5 Calculate Random Standard Uncertainty

The random standard uncertainty of the meanvalue at each traverse point presented in Table10-1.2 is calculated from eq. (4-3.3) as

45


SXijp

SXij

�Nij

where values for SXijare shown in Table 10-1.3-1.

Since there are 60 measurements at each traversepoint, Nij p 60, the degrees of freedom at eachtraverse point is

�ij p Nij − 1 p 59 (10-1.3)

The resulting values for SXijare also presented

in Table 10-1.3-1.

10-1.6 Propagate Random Standard Uncertainty

The random standard uncertainty for each aver-age velocity along a radius SVi

is calculated fromeq. (7-3.1) as

SVi p ��10

jp1��∂Vi/∂Xij��sXij

��21⁄2

(10-1.4)

where

∂Vi

∂Xijp

110

The random standard uncertainty of the averagevelocity in the pipe sV is then calculated as

sV p �4

jp1 � ∂V

∂VisVi�2

1 ⁄2

(10-1.5)



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Table 10-1.6 Standard Deviation of AverageVelocity (Example 10-1)

Parameter, ft/sec Value

sV1

0.0440sV

20.0523

sV3

0.0618sV

40.0687

sV 0.0287

where

∂V

∂Vip

14

The results are summarized in Table 10-1.6.While not shown here, the degrees of freedomassociated with each sVi

as well as SV can becalculated using eq. (B-1.7).

10-1.7 Calculate Systematic StandardUncertainties

10-1.7.1 Calibration. Velocity is measured ateach point with the same Pitot tube, digital pres-sure transmitter, and DAS. The Pitot tube, digitalpressure transmitter, and DAS are calibrated to-gether as a system. For this example, the calibrationuncertainty of the instruments, estimated as thelimits of the elemental systematic error at 95%confidence, is 3% of measured velocity. Using eq.(4-3.4), the elemental systematic standard uncer-tainty associated with the calibration of the instru-ments, of the average pipe velocity measurementis estimated as

bVcp (0.03/2)(5.57) p 0.0836 (ft/sec) (10-1.6)

The degrees of freedom of bVCis assumed to

be large (≥30).

10-1.7.2 Spatial Variation. The true value beingmeasured is the integrated average of the velocityover the cross-sectional flow area. Averaging thevelocities at the centroid of equal area pointsis a numerical approximation of this integratedaverage. Even if exact values of velocity are knownat each of the traverse points, the analyst mustrecognize that the numerical average may notequal the integrated-average value over the entirecross-sectional flow area. This is due to incompletesampling of a profile which varies as a function

46


of spatial position. Therefore, there is an inherentuncertainty in the method used to approximatethe integrated-average velocity. This uncertaintyof method is sometimes referred to as uncertaintydue to spatial variation. Since the velocity profileremains fixed for the duration of the test, the spatialvariation is a source of systematic uncertainty forthe test result.

This uncertainty can be estimated in a varietyof ways, including

(a) special tests which provide independentknowledge of the velocity profile;

(b) special tests which compare the measurementtechnique to other techniques which yield the de-sired integrated-average;

(c) published reports which document the uncer-tainty of similar measurement techniques in similarmeasurement situations; and

(d) evaluation of the variation in test measure-ments as a function of spatial position.

For this example, the velocity profile is distorteddue to the presence of flow disturbances upstreamof the measurement location. Special tests hadpreviously been conducted to determine if takingadditional traverse points (20) along each of the4 radii would result in a significant change in themeasurement of the overall average velocity inthe pipe, as computed using eq. (10-1.2). The resultsof these past tests were used to compute a standarddeviation representing the dispersion of errors(differences) in average pipe velocities computedusing 20 traverse points along each of the 4 radiiversus those obtained using 10 traverse points.The standard deviation published from these pasttests was assumed to be representative of thesystematic standard uncertainty (for the presenttest) resulting from spatial variation in the radialdirection. The resulting value is shown in thefollowing equation. The published degrees of free-dom for this value is 30.

bVSRp 0.0111 ft/sec

Since no additional pipe taps are available fortesting, it is unknown whether the average of thetraverses of the four radii sufficiently characterizesthe integrated average around the pipe due tocircumferential variations. Therefore, the uncer-tainty due to spatial variation must be inferredfrom available data. For this example, it will beestimated by evaluation of the variation in theradial averages. Assuming that the four measured



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


radial averages come from a population of valueswhich are normally distributed, the systematicstandard uncertainty in the average velocity mea-surement due to spatial variation in the circumfer-ential direction is estimated using eq. (4-3.5) as

bVSCp � 1

N� Vi

�NVi

i p 1

(Vi − V)2

NVi− 1

p 0.0494 ft/sec (10-1.7)

where

NVip the number of independent radial average

velocities used in the computation ofbVSC

N�Vi

p the number of radial average velocitiesaveraged together in the computation ofV. Both of these values are equal to 4.The degrees of freedom associated withbVSC

is equal to NVi− 1 p 3.

10-1.8 Propagate Systematic StandardUncertainties

The systematic standard uncertainties due tocalibration and spatial variation are combined us-ing eq. (7-4.1) as follows:

bV p [(bVC)2 +(bVSR

)2 + (bVSC)2]

1⁄2

p [(0.0836)2 + (0.0111)2 + (0.0494)2]1⁄2 (10-1.8)

p 0.0977 (ft/sec)

10-1.9 Uncertainty of Result

The combined standard uncertainty of the re-sulting average pipe velocity measurement is deter-mined using eq. (7-5.1) as follows:

uV p [(bV)2 + (sV)2]1 ⁄2 p 0.102 (ft/sec) (10-1.9)

The expanded uncertainty of the resulting aver-age pipe velocity measurement is determined usingeq. (7-5.2) as follows:

UV p 2uV p 0.204 (ft/sec)

Use of the 2 in the above equation is appropriateas it can be shown using eq. (B-1.7) that thecombined degrees of freedom of the result is≥30. A summary presentation of the uncertainty

47


analysis at the test result level is reported in Table10-1-9.

10-2 FLOW RATE UNCERTAINTY [12]

10-2.1 General Description

In this example, the test objective is to determinethe flow of water using a 6 x 4 in. venturi (seeFig. 10-2.1) within an uncertainty of 0.5%. A pretestanalysis is required to determine if an uncalibratedventuri could be used to satisfy the test objective,and, if not, whether calibration of the venturiwould achieve the desired objective. This exampleoutlines differences in the analysis for both theuncalibrated and calibrated cases. The examplelooks at each case in both absolute and relative(i.e., percent) value formats.

The clearest way to present the results of thesteps in the uncertainty analysis is to developa table in which the names, definitions, values,uncertainties, sensitivities, etc., are displayed foreach of the variables required for the analysis.The table associated with the uncalibrated case isshown here first, as Table 10-2.1-1. The table hasbeen developed in accordance with the step-by-step procedure of Section 9. The steps in thedevelopment of the table are as follows:

(a) Define Measurement Process. Flow rate can becalculated by making the measurements requiredto define the independent variables found in eq. (10-2.1)[21].

m p0.099702Cd2 ��h

�1 − � dD�

4(10-2.1)

The definitions and values for each of the mea-surements used in the calculation of the mass flowrate are displayed in the first four columns ofTable 10-2.1-1.

10-2.2 Uncalibrated Venturi Case

(b), (c), (d) List Elemental Systematic UncertaintySources; Estimate Elemental Uncertainties; Calculate theSystematic and Random Uncertainties. The systematicvalues were evaluated and are listed in Table 10-2.1-1. The degrees of freedom associated with eachof the uncertainty estimates are assumed to begreater than 30. The systematic uncertainties foreach input parameter are displayed on an absolutebasis. The standard deviation of the mean of the



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Table 10-1.9 Uncertainty of Result (Example 10-1)

Absolute Absolute AbsoluteSystematic Random Combined AbsoluteStandard Standard Standard Expanded

Uncertainty, Uncertainty, Uncertainty, Uncertainty,Symbol Description Units Calculated Value bv sv uv Uv

V Average ft/sec 5.57 0.0977 0.0287 0.102 0.204velocity inpipe

Fig. 10-2.1 Schematic of a 6 in. x 4 in. Venturi

Table 10-2.1-1 Uncalibrated Case (Example 10-2)


RandomAbsolute Standard

Systematic UncertaintyStandard of the

Nominal Uncertainty, Mean,Symbol Description Units Value bX sX

i

C Discharge coefficient . . . 0.984 3.75E–03 0d Throat diameter in. 3.999 5.0E–04 0D Inlet diameter in. 6.001 1.0E–03 0� Water density at 60°F [25] lbm/ft3 62.37 0.002 0.002h Differential pressure head in. H2O 100 0.15 0.4

across venturi (at 68°F)

GENERAL NOTES:(a) The systematic and random estimates for density are based on water temperature measurements

having systematic and random uncertainties of 0.2°F and 0.1°F, respectively.(b) The systematic uncertainty for the differential pressure head is assumed to be one-half the least

count of the manometer scale.

48




--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


measurements taken is shown for each of the inde-pendent variables. These values are also expressedin absolute terms, so the units are the same as theunits for the measured parameter.

(e) Propagate the Systematic and Random Uncertain-ties. The sensitivity of the result to each of the indi-vidual parameter uncertainties is calculated, eithernumerically or analytically, in accordance with sub-section 7-2.

A quick and easy way to numerically calculatethe sensitivity coefficients of the independentparameters is to develop a table using a spread-sheet program on the personal computer. Table10-2.1-2 shows the results obtained using such aspreadsheet, along with the formulas used in thespreadsheet.

As can be seen by looking at the formulas shownin Table 10-2.1-2, the same basic equation for thecalculation of m was repeated seven times (oncefor each independent parameter, X, with Xi beingreplaced with Xi + XiQx, where Qx is the quantityby which Xi is to be perturbed. A forward differenc-ing scheme is used with a value of Qx equalto 0.1% (i.e., 0.001). A perturbed m is thereforecalculated for each independent parameter andthe baseline value of m is subtracted from eachperturbed m. Dividing the difference by the amountby which Xi was perturbed (Xi + XiQx) providesthe absolute sensitivity coefficient.

As noted in para. 7-2.1, the sensitivity coefficientcan also be determined analytically, by findingthe partial derivatives of the result with respectto each of the measured parameters. Table 10-2.1-3 shows the formulas for the partial derivativesfor each of the measured parameters with respectto the calculated mass flow rate, and the sensitivitycoefficients found using these formulas. By lookingat the numbers in the last two columns of Table10-2.1-4, it can be seen at once which parameterscontribute most to the uncertainty of the result.In this case, the largest contributor is the systematicuncertainty in the discharge coefficient, C.

(f) Calculate Uncertainty. The total uncertainty ofthe result is then calculated by root-sum-squaringthe systematic and random contributions. Table 10-2.1-5 shows the nominal value, and the systematic,random, and total uncertainties for m, calculated asdescribed previously.

10-2.2.1 Relative Sensitivities and Uncertain-ties. Sensitivities and uncertainties may also bestated in relative terms. To convert the uncertaintyof any parameter from absolute to relative terms,

49


it is necessary only to divide the absolute uncer-tainty by the nominal value of the parameter.Multiplying by 100 would provide an expressionof the uncertainty in terms of a percentage of thenominal value.

The sensitivity coefficients were converted torelative terms using eq. (7-2.2). Multiplying by 100yields the percentage change in the result for a1% change in the measured parameter. Tables 10-2.1.1-1, 10-2.1.1-2, and 10-2.1.1-3 display the sameparameters as Tables 10-2.1-1, 10-2.1-4, and 10-2.1-5, respectively, but with the sensitivities anduncertainties expressed in relative (percent) termsrather than in absolute terms.

10-2.3 Calibrated Venturi Case

(a) Define the Measurement Process(b), (c), (d) List Elemental Uncertainty Sources; Esti-

mate Elemental Uncertainties; Calculate the Systematicand Random Uncertainties. In this case the systematicuncertainty of the discharge coefficient changes. Inaddition, the systematic uncertainties of the throatand pipe diameters are eliminated (set to zero inTable 10-2.1.1-4) because the entire flow section iscalibrated as a unit, which is standard practice. Theuncertainty estimate for discharge coefficient comesfrom calibration data, which has been fossilized, andis listed as a systematic uncertainty in Table 10-2.1.1-4 as Bc/C p 0.122%, which corresponds to anabsolute systematic standard uncertainty ofbc p Bc/2 p 0.061%.

(e), (f) Propagate the Systematic and Random Uncer-tainties; Calculate Uncertainty. These new valuescan be inserted into the tables (spreadsheets) usedfor the uncalibrated case. As the formulas do notchange, there is nothing more that needs to bedone. The parameters and results are displayedusing a relative basis for the uncertainties andsensitivities in Tables 10-2.1.1-4, Table 10-2.1.1-5,and 10-2.1.1-6. A summary of these uncertaintiesis given in Table 10-2.1.1-7.

The test objective of m ± 0.5% is marginallysatisfied by using a calibrated venturi (Table 10-2.1-11). The most promising path to take to obtainingadditional reductions in the uncertainty of themass flow rate determination would be throughan increase in the number of differential headreadings to reduce sX.

(g) Report. Tables 10-2.1.1-1 through 10-2.1.1-7would be included in the report, with text notingthat calibration of the venturi is required to meet



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Table 10-2.1-2 Absolute Sensitivity Coefficients in Example 10-2 (Calculated Numerically)

Quantity by Which to Perturb Independent Parameters (Qx): 0.1%

Formulas for Absolute Sensitivity Absolute Sensitivity

Symbol, Nominal �m�Xi

�m�XiXi Values

C 0.984 140.60.099702 (C + CQx) d2 ��h

�1 − �dD�

4− m

CQx

d 3.999 86.30.099702(d + dQx) d2 ��h

�1 − �d + dQx

D �4

− m

dQx

D 6.001 −11.30.099702d2 ��h

�1 − � dD + DQx�

4− m

DQx

� 62.37 1.110.099702d2 �(� + �Qx)h

�1 − �dD�

4− m

�Qx

h 100 0.6920.099702d2 ��(h + hQx)

�1 − �dD�

4− m

hQx

Symbol Result Formula for m

m 138.4 0.099702Cd2 ��h

�1 − �dD�

4

the objective of 0.5% uncertainty or less in themass flow rate determination.

10-3 FLOW RATE UNCERTAINTY INCLUDINGNONSYMMETRICAL SYSTEMATIC STANDARDUNCERTAINTY

This example is nearly identical to that presentedin subsection 10-2 for the uncalibrated venturicase. The only differences are the following:

50


(a) the discharge coefficient will have a nonsym-metrical absolute systematic uncertainty (95% confi-dence estimate) represented by B− p 0.0095 andB+ p 0.0055; and

(b) the differential pressure head across the ven-turi (at 68°F) will have a nonsymmetrical absolutesystematic uncertainty (95% confidence estimate)represented by B− p 0.1 in. H2O and B+ p 0.5 in.H2O.

Using the methodology in subsection 8-2, thedischarge coefficient will have a nonsymmetrical



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Table 10-2.1-3 Absolute Sensitivity Coefficients in Example 10-2 (Calculated Analytically)

AbsoluteFormulas for Absolute Sensitivity Sensitivity

Symbol, Nominal �m�Xi

�m�XiXi Values

C 0.984 1400.099702d2 ��h

�1 − �dD�

4

�1 − �dD�

4

(2)0.099702C ��hd − ��0.099702C��h� d2 (−4) �dD�

3

2 �1 − �dD�

4

D�1 − �dD�

4

� �d 3.999 72.6

−4(0.099702)Cd6 ��h

2D5 �1 − �dD�

4

�3⁄2D 6.001 −11.4

0.099702Cd2 �h/�

2

�1 − �dD�

4� 62.37 1.11

0.099702Cd2 ��/h

2

�1 − �dD�

4h 100 0.692

absolute systematic standard uncertainty of b− p0.00575 and b+ p 0.00175, and the differentialpressure head across the venturi will have a non-symmetrical absolute systematic standard uncer-tainty of b− p −0.05 in. H2O and b+ p 0.35 in.H2O. The results of this uncertainty analysis arepresented in Tables 10-3-1 and 10-3-2 in whicheach symbol has the same description as in Table10-2.1-1, and in which each corresponding numberhas the same units as those given in Table 10-2.1-1.

Following the method given in subsection 8-2,the absolute systematic standard uncertainty, b,for each variable is estimated by b p (b+ + b−)/2 and the offset, q, is estimated by q p (b+ − b−)/2. The absolute sensitivity, �, for each variable isestimated with all variables set to their offsetvalues, X + q. The apparent offset in the mass

51


flow rate, qr, is determined by the difference be-tween the mass flow rate evaluated when allvariables are set to their offset values, RX + q, andthe mass flow rate evaluated when all variablesare set to their measured values, RX. The lowerand upper limits on the uncertainty of the massflow rate are given by U− p U95,R − qr, and U+ pU95,R + qr, respectively.

10-4 COMPRESSOR PERFORMANCE UNCERTAINTY

The following example follows the step-by-stepprocedure outlined in Section 9. This examplehighlights the following:

(a) the identification and quantification of ele-mental sources of uncertainty;



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Table 10-2.1-4 Absolute Contributions of Uncertainties of Independent Parameters (Example 10-2:Uncalibrated Case)



(in Parameter Units) (in Results Units Squared)



Symbol Nominal Uncertainty, Uncertainty, Sensitivity, Contribution, Contribution,Xi Description Units Value bX sX

i�i (bX

i�i )

2 (sXi�i )

2

C Discharge coefficient . . . 0.984 3.75E–03 0 140 0.276 0d Throat diameter in. 3.999 5.0E–04 0 86.2 1.86 � 10−3 0D Inlet diameter in. 6.001 1.0E–03 0 −11.4 1.29 � 10−4 0� Water density at lbm/ft3 62.37 0.002 0.002 1.11 4.93 � 10−6 4.92 � 10−6

60°F [25]h Differential pressure in. H2O 100 0.15 0.4 0.692 1.08 � 10−2 7.66 � 10−2


Table 10-2.1-5 Summary: Uncertainties in Absolute Terms (Example 10-2: Uncalibrated Case)

CombinedAbsolute StandardRandom Uncertainty Total

Absolute Systematic Standard of the AbsoluteCalculated Standard Uncertainty, Uncertainty, Result, Uncertainty,

Symbol Description Units Value bR sR uR UR,95

m Mass flow rate lbm/sec 138.4 0.537 0.276 0.604 1.21

Table 10-2.1.1-1 Relative Uncertainty of Measurement (Example 10-2:Uncalibrated Case)


Relative RelativeSystematic RandomStandard Standard

Nominal Uncertainty, Uncertainty,Symbol Description Units Value b′X

i� bX

i/Xi s′X

i� sX

i/Xi

C Discharge coefficient . . . 0.984 0.361% 0%d Throat diameter in. 3.999 0.0125% 0%D Inlet diameter in. 6.001 0.0167% 0%� Water density at 60°F [24] lbm/ft3 62.37 0.0032% 0.0032%h Differential pressure head in. H2O 100 0.150% 0.400%


52




--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Table 10-2.1.1-2 Relative Contributions of Uncertainties of Independent Parameters (Example 10-2:Uncalibrated Case)


RelativeSystematic RelativeStandard Random

Relative Uncertainty UncertaintyNominal Sensitivity, Contribution, Contribution,

Symbol Description Units Value �i′ (b′Xi�i′)2 (s′X

i�i′)2

C Discharge coefficient . . . 0.984 1.0 1.45 � 10−5 0d Throat diameter in. 3.999 2.4923 9.71 � 10−8 0D Inlet diameter in. 6.001 −0.4923 6.73 � 10−9 0� Water density at 60°F [24] lbm/ft3 62.37 0.50 2.57 � 10−10 2.57 � 10−10

h Differential pressure head in. H2O 100 0.50 5.62 � 10−7 4.0 � 10−6


Table 10-2.1.1-3 Summary: Uncertainties in Relative Terms for the Uncalibrated Case

Calculated Result

RelativeRelative Relative Combined Relative

Systematic Random Standard ExpandedCalculated Uncertainty, Uncertainty, Uncertainty, Uncertainty,

Symbol Description Units Value bR/R sR/R uR/R UR,95/R

m Mass flow rate lbm/sec 138.4 0.39% 0.20% 0.44% 0.88%

Table 10-2.1.1-4 Relative Uncertainties of Independent Parameters(Example 10-2: Calibrated Case)



Nominal Uncertainty, Uncertainty,Symbol Description Units Value b′X

i� bX

i/Xi s′X

i� sX

i/Xi

C Discharge coefficient . . . 0.984 0.061% 0%d Throat diameter in. 3.999 0% 0%D Inlet diameter in. 6.001 0% 0%� Water density at 60°F [24] lbm/ft3 62.37 0.0032% 0.0032%h Differential pressure head in. H2O 100 0.150% 0.400%


53




--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Table 10-2.1.1-5 Relative Contributions of Uncertainties of Independent Parameters (Example 10-2:Calibrated Case)



Relative Uncertainty UncertaintyNominal Sensitivity, Contribution, Contribution,

Symbol Description Units Value �i′ (b′Xi� ′i )2 (s′X

i� ′i )2

C Discharge coefficient . . . 0.984 1.0 3.72 � 10−7 0d Throat diameter in. 3.999 2.4923 0 0D Inlet diameter in. 6.001 −0.4923 0 0� Water density at 60°F [24] lbm/ft3 62.37 0.50 2.57 � 10−10 2.57 � 10−10

h Differential pressure head in. H2O 100 0.50 5.62 � 10−7 4.0 � 10−6


Table 10-2.1.1-6 Summary: Uncertainties in Relative Terms for the Calibrated Case

Calculated Results

RelativeRelative Relative Combined Relative

Systematic Random Standard ExpandedCalculated Uncertainty, Uncertainty, Uncertainty, Uncertainty,

Symbol Description Units Value bR/R sR/R uR/R UR,95/R

m Mass flow rate lbm/sec 138.4 0.097% 0.20% 0.22% 0.44%

Table 10-2.1.1-7 Summary: Comparison BetweenCalibrated and Uncalibrated Cases

Calibrated Case Uncalibrated Case

m 138.4 lbm/sec 138.4 lbm/secbR/R 0.097% 0.39%sR/R 0.20% 0.20%uR/R 0.22% 0.44%

UR,95/R 0.44% 0.88%

(b) treatment of correlated sources of uncertaintyin a practical manner; and

(c) the importance of applying all knownengineering corrections and using appropriate engi-neering relationships as part of the test results analy-sis process.

10-4.1 Define the Measurement Process

A test was conducted to determine the adiabaticefficiency of an air compressor at normal operating

54


conditions. The compressor was operated at nor-mal, steady-state conditions for two hours priorto the test and for one hour during the test.

Measurements of the total pressure and totaltemperature at the inlet and exit of the compressorwere collected at one minute intervals resultingin 60 discrete measurements of each parameterover the test period. The mean value for eachmeasured parameter was calculated using eq.(4-3.1). The resulting averages are presented inTable 10-4.1-1. A simplified schematic depictingthe test measurement locations is shown in Fig.10-4.1.

The compressor inlet and exit total pressureswere measured using multiport impact pressurearrays which are permanently installed in thecompressor inlet and exit. Each array was con-nected to a digital pressure indicator in a mannerwhich yielded a spatially averaged pressure mea-surement. The compressor inlet and exit total tem-peratures were measured using multipoint thermo-couple stagnation probe arrays which werepermanently installed in the compressor inlet and



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Table 10-3-1 Absolute Contributions of Uncertainties of Independent Parameters (Example 10-3:Uncalibrated, Nonsymmetrical Systematic Uncertainty Case)


Parameter Information(in Parameter Units)

Absolute AbsoluteNegative Positive Absolute Absolute

Systematic Systematic Systematic RandomStandard Standard Standard Standard

Symbol Nominal Uncertainty, Uncertainty, Uncertainty, Uncertainty,Xi Description Units Value bX

− bX+ bX s X

i

C Discharge coefficient . . . 0.984 5.75E–03 1.75E–03 3.75E–03 0d Throat diameter in. 3.999 5.0E–04 5.0E–04 5.0E–04 0D Inlet diameter in. 6.001 1.0E–03 1.0E–03 1.0E–03 0� Water density at lbm/ft3 62.37 0.002 0.002 0.002 0.002

60°F [24]h Differential pressure in. H2O 100 −0.05 0.35 0.15 0.4



Uncertainty Contribution ofParameters to the Result(in Result Units Squared)

Absolute AbsoluteSystematic RandomStandard Standard

Absolute Uncertainty UncertaintySymbol, Sensitivity, Contribution, Contribution,

Xi Description Units �i (bXi�i)

2 (sXi�i)

2

C Discharge coefficient . . . 140.8 0.279 0d Throat diameter in. 86.14 1.86 � 10−3 0D Inlet diameter in. −11.34 1.29 � 10−4 0� Water density at 60°F lbm/ft3 1.108 4.9 � 10−6 4.9 � 10−6

[24]h Differential pressure head in. H2O 0.6898 1.07 � 10−2 7.61 � 10−2


exit. The thermocouples from each multiprobearray were connected in parallel in a mannerwhich yielded a spatially averaged temperaturemeasurement. A common digital temperature indi-cator, with built-in cold junction compensation,was used to measure both the inlet and the exittemperatures. Pretest and posttest calibrations wereperformed on all instrumentation.

The adiabatic efficiency of the air compressorwas initially calculated using the following simpli-fied engineering relationship:

55


� p(P2/P1)[(�−1)/�] − 1

T2/T1 − 1(10-4.1)

where

P1p measured compressor inlet total pressureP2p measured compressor exit total pressureT1p measured compressor inlet total temper-

atureT2p measured compressor exit total temper-

ature



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Table 10-3-2 Summary: Uncertainties in Absolute Terms (Example 10-3: Uncalibrated, NonsymmetricalSystematic Uncertainty Case)

CombinedAbsolute Absolute Standard

Systematic Random Uncertainty ExpandedStandard Standard of the Absolute

Calculated Uncertainty, Uncertainty, Result, Uncertainty,Symbol Description Units Value bR sR uR UR,95

m Mass flow rate lbm/sec 138.4 0.540 0.276 0.606 1.21

Expanded ExpandedAbsolute AbsoluteNegative Positive

Offset, Uncertainty, Uncertainty,Symbol Description Units qR UR,95

− UR,95+

m Mass flow rate lbm/sec −0.14 1.35 1.07

Table 10-4.1-1 Elemental Random StandardUncertainties Associated With Error Sources

Identified in Para. 10-4.2

Absolute RandomStandard

Mean Value, Uncertainty,Symbol Description Units Xi sX

i

P1 Inlet pressure psia 14.70 0.030P2 Exit pressure psia 95.52 0.170T1 Inlet temperature R 520.0 0.300T2 Exit temperature R 960.0 0.600

�p ratio of the specific heats, assumed tobe 1.40

�p adiabatic compressor efficiency

Simplifying assumptions for this relationshipinclude

(a) use of ideal gas properties for dry air(b) negligible potential energy change(c) negligible heat loss to surroundings(d) constant specific heats and a specific heat ratio

of 1.4Using the previous simplified engineering rela-

tionship, a test result and associated uncertaintywere calculated and are presented in Tables 10-4.1-2 and 10-4.1-3. The details of the uncertaintyanalysis are discussed in paras. 10-4.2 through 10-4.6. Recognizing that the use of the simplifiedengineering relationship in the computation of

56


the test result introduces an error that was notaccounted for in the reported uncertainty, thetest result was reevaluated using more rigorousengineering relationships to eliminate some of theassumptions noted. The difference between thetwo test results was compared to the previouslyreported uncertainty to illustrate the importanceof applying all known engineering corrections andusing appropriate engineering relationships as partof the results analysis process. The details of thiscomparison are discussed in para. 10-4.7.

10-4.2 List Elemental Error Sources

Based upon a review of the measurement meth-ods and instruments employed for the test, thefollowing lists of elemental error sources werecompiled for each of the measurements:

(a) Compressor Inlet and Exit Pressure Measurements(1) random error associated with incomplete

sampling of the average pressure over the durationof the test and random variability in the pressuremeasurement instrumentation

(2) systematic error resulting from imperfectcalibration and drift of the digital pressure indicator

(3) systematic error resulting from environ-mental influences on the digital pressure indicator

(4) systematic error resulting from imperfectspatial averaging

(5) systematic error due to the inability of theimpact pressure arrays to fully realize total pressure



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Fig. 10-4.1 Typical Pressure and Temperature Locations for Compressor Efficiency Determination

Table 10-4.1-2 Independent Parameters

Uncertainty Contribution ofParameters to the Result

Parameter Information (in Parameter Units) (in Result Units2)


Systematic Random Standard StandardNominal Standard Standard Absolute Uncertainty UncertaintyValue, Uncertainty, Uncertainty, Sensitivity, Contribution, Contribution,

Symbol Description Units Xi b Xi

s Xi

�i (�i bXi)2 (�is

Xi

)2

P1 Inlet pressure psia 14.70 0.021 0.030 −0.0409 7.53E–7 1.51E–6P2 Exit pressure psia 95.52 0.198 0.170 +0.00629 1.54E–6 1.14E–6T1 Inlet temperature R 520.0 0.646 [Note (1)] 0.300 +0.00367 5.62E–6 [Note (1)] 1.21E–6T2 Exit temperature R 960.0 0.792 [Note (1)] 0.600 −0.00203 2.59E–6 [Note (1)] 1.48E–6

NOTE:(1) These systematic standard uncertainties have some components that are correlated. The correlated terms are not shown in the table.

Table 10-4.1-3 Calculated Result

Absolute Absolute AbsoluteSystematic Random Combined AbsoluteStandard Standard Standard Expanded

Calculated Uncertainty, Uncertainty, Uncertainty, Uncertainty,Symbol Description Units Value, R b R sR uR UR

� Computed adiabatic Nondimensional 0.8355 0.00309 0.00231 0.00366 0.00772compressor efficiency

57




--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Table 10-4.3.2-1 Inlet and Exit Pressure Elemental Systematic StandardUncertainties

Absolute AbsoluteSystematic Systematic

Limits of Standard Limits of StandardError for Uncertainty Error for Uncertainty

Source of Information for Inlet for Inlet Exit for ExitEstimation of Limits of [Note (1)] [Note (2)] [Note (1)] [Note (2)]

Error Source Error psia, BP1 psia, bP1 psia, BP2 psia, bP2

Calibration and Calibration and drift 0.04 0.02 0.25 0.125drift of indicator uncertainty reported[Note (3)] by calibration

laboratory based uponpretest and posttestcalibration

Environmental Published information 0.01 0.005 0.06 0.03influences on provided by indicatorindicator vendor[Note (3)]

Spatial averaging Reported uncertainty 0.01 0.005 0.3 0.15[Note (3)] provided by

compressor vendorbased upon test rigdata

Realization of Engineering judgment Negligible Negligible Negligible Negligibletotal pressure[Note (3)]

Total Eq. 6.8 N/A 0.021 N/A 0.198

NOTES:(1) All limits of error are estimated at 95% confidence. It is assumed that these estimates are based on

large degrees of freedom and that the population of possible error values associated with eachelemental systematic error source is normally distributed.

(2) Elemental systematic standard uncertainties are calculated using eq. (4-3.5).(3) It is assumed that the indicated elemental systematic standard uncertainties for the inlet and exit

pressure measurements are not correlated because the inlet and exit measurements are substantiallydifferent in magnitude. For this particular example, assuming that these are not correlated will elevatethe estimated uncertainty in the test result.

(b) Compressor Inlet and Exit Temperature Measure-ments

(1) random error associated with incompletesampling of the average temperature over the dura-tion of the test and random variability in the temper-ature measurement instrumentation

(2) systematic error resulting from imperfectcalibration and drift of the thermocouple probes

(3) systematic error resulting from imperfectcalibration and drift of the digital temperature indi-cator

(4) systematic error resulting from imperfectcalibration and drift of the cold junction reference

(5) systematic error resulting from environ-mental influences on the digital temperature indi-cator

(6) systematic error resulting from imperfectspatial averaging

58


(7) systematic error due to the inability of thestagnation probes to fully realize total temperature

10-4.3 Calculate Random and SystematicStandard Uncertainties

10-4.3.1 Random Standard Uncertainty. The ele-mental random standard uncertainties associatedwith the error sources identified in para. 10-4.2were evaluated by calculating the absolute stan-dard deviation of the mean for each measuredparameter [see eq. (4-3.2)]. The results are summa-rized in Table 10-4.1-1.

10-4.3.2 Systematic Standard Uncertainty. Theelemental systematic standard uncertainties associ-ated with the error sources identified in para. 10-4.2 were evaluated and totaled for each measured



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Table 10-4.3.2-2 Inlet and Exit Temperature Elemental Systematic StandardUncertainties

Absolute AbsoluteSystematic Systematic

Limits of Standard Limits of StandardError for Uncertainty Error for Uncertainty

Source of Information for Inlet for Inlet Exit for ExitEstimation of Limits of [Note (1)] [Note (2)] [Note (1)] [Note (2)]

Error Source Error °R, BT1 °R, bT1 °R, BT2 °R, bT2

Calibration and Calibration and drift 1.1 0.55 1.1 0.55drift of uncertainty reported bythermocouple calibration laboratoryprobes based upon pretest and[Note (3)] posttest calibration

Calibration and Calibration and drift 0.4 0.2 0.2 0.1drift of indicator uncertainty reported by[Note (3)] calibration laboratory

based upon pretest andposttest calibration

Calibration and Calibration and drift 0.5 0.25 0.5 0.25drift of cold uncertainty reported byjunction calibration laboratoryreference based upon pretest and[Note (4)] posttest calibration

Environmental Published information 0.2 0.1 0.1 0.05influences on provided by indicatorindicator vendor[Note (3)]

Spatial averaging Reported uncertainty 0.1 0.05 1.0 0.5[Note (3)] provided by compressor

vendor based upon testrig data

Realization of Engineering judgment Negligible Negligible Negligible Negligibletotaltemperature[Note (3)]

Total Eq. (6-2.1) N/A 0.0646 N/A 0.792

NOTES:(1) All limits of error are estimated at 95% confidence. It is assumed that these estimates are based on

large degrees of freedom and that the population of possible error values associated with eachelemental systematic error source is normally distributed.

(2) Elemental systematic standard uncertainties are calculated using eq. (4-3.4).(3) It is assumed that the indicated elemental systematic standard uncertainties for the inlet and exit

temperature measurements are not correlated because the inlet and exit measurements aresubstantially different in magnitude. For this particular example, assuming that these are not correlatedwill elevate the estimated uncertainty in the test result.

(4) It is assumed that the indicated elemental systematic standard uncertainties for the inlet and exittemperature measurements are correlated as an error in the common cold junction reference willcause equivalent errors in the inlet and exit measurements.

parameter as shown in Tables 10-4.3.2-1 and 10-4.3.2-2.

10-4.4 Propagate Random and SystematicStandard Uncertainties

The individual parameter standard uncertaintiesare propagated into terms of the test result by aTaylor series expansion as given in Nonmandatory

59


Appendix C. The absolute random standard uncer-tainty for the test result is [see eq. (7-3.1)]

sR p [(�P1sP1

)2 + (�P2sP2

)2 + (�T1sT1

)2 + (�T2sT2

)2]1 ⁄2

p 0.00231 (10-4.2)

where the absolute sensitivities are [see eq. (7-2.1)]



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


�P1p

−[� − 1)/�][(P2/P1)−1/�][P2/(P1)2][(T2/T1) − 1]

(10-4.3)

p −0.0409(psia−1)

�P2p

[(� − 1)/�][(P2/P1)−1/�][1/P1][(T2/T1) − 1]

(10-4.4)

p +0.00629(psia−1)

�T1p

[T2]{[(P2/P1)(�−1)/�] − 1}[(T2 − T1)2]

(10-4.5)

p +0.00367(°R−1)

�T2p

[−T1]{[(P2/P1)(�−1)/�] − 1}[(T2 − T1)2]

(10-4.6)

p −0.00203(°R−1)

Note: In computing the preceeding sensitivity coefficients, itwas assumed that the specific heat ratio is independent of airtemperature.

The absolute systematic standard uncertainty forthe test result is [see eq. (8-1.2)]

bR p [(�P1bP1

)2 + (�P2bP2

)2 + (�T1bT1

)2

+ (�T2bT2

)2 +2�T1�T2

bT1T2]

1⁄2 p 0.00309 (10-4.7)

where

bT1T2p the covariance of the error sources com-

mon to T1 and T2 and is determined as(see subsection 8-1)

bT1T2p 0.25(°R) · 0.25(°R) p 0.0625(°R2) (10-4.8)

10-4.5 Calculate Uncertainty

The combined standard uncertainty in the testresult is [see eq. (7-5.1)]

uR p ��bR�2 + �SR�2�1⁄2

p 0.00386 (10-4.9)

and the expanded uncertainty of the result is [seeeq. (7-5.2)]

UR p 2uR p 0.00772 (10-4.10)

The assumptions required for using this equationare presented in subsection 1-3.

60


The uncertainty interval for the test result is[see eq. (7-5.3)]

R ± UR p 0.8355 ± 0.00772 (10-4.11)

10-4.6 Report

A summary presentation of the uncertainty anal-ysis at the test result level is reported in Tables10-4.1-2 and 10-4.1-3.

10-4.7 Importance of Applying KnownEngineering Corrections

As discussed in subsection 4-1, it is importantthat all known engineering corrections are appliedand appropriate engineering relationships are usedas part of the results analysis process. Failure todo so leads to additional errors in the reportedtest result which are not accounted for in the testuncertainty analysis. This could, in turn, lead toexpression of an uncertainty interval for a testresult that does not encompass the true value.

For this particular example, the assumptions ofthe analysis method used to compute the adiabaticefficiency of the compressor were not identifiedas sources of error. To illustrate the potentialsignificance of having overlooked these sources oferror, the test result will be recalculated usingmore exact relationships that do not require theassumption of constant specific heats and the as-sumed specific heat ratio of 1.4.

By definition, the adiabatic efficiency of a com-pressor is the ratio of the work input required toraise the pressure of a gas to a specified value inan isentropic manner to the actual work input.

� pws

wa(10-4.12)

where

wsp entropic compressor workwap actual compressor work

Assuming negligible change in the potential en-ergy of the gas being compressed and negligibleheat loss to the surroundings, the adiabatic effi-ciency can be expressed as a function of stagnationenthalpies as follows:

� ph2s − h1

h2 − h1(10-4.13)



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Fig. 10-4.7 The h-s Diagram of the Actual and Isentropic Processes of an Adiabatic Compressor(Used by Permission From McGraw-Hill Co.)

where

h1p stagnation enthalpy at measured inlet con-ditions

h2p stagnation enthalpy at measured exit con-ditions

h2sp stagnation enthalpy assuming an isen-tropic compression process

An h-s diagram of the actual and isentropicprocesses of an adiabatic compressor is illustratedin Fig. 10-4.7.

Assuming ideal gas properties for dry air, idealgas property tables for air are used to evaluateh1, h2, and h2s. The values for h1 and h2 are evaluatedat the measured inlet and exit conditions. Thevalue for h2s, however, must be evaluated at astate corresponding to the measured exit pressurebut assuming isentropic compression has occurred.This is done by first using the following relation-ship, which accounts for variable specific heats,to determine the relative pressure correspondingto isentropic compression.

Pr2 p Pr1 �P2

P1�s p const(10-4.14)

61


where

P1p measured compressor inlet total pressureP2p measured compressor exit total pressure

Pr1p the relative pressure determined fromideal gas property tables for air at themeasured compressor inlet total temper-ature

Pr2p the relative pressure at the exit corres-ponding to isentropic compression

Next, the value for h2s is evaluated from airproperty tables at a state corresponding to thevalue of Pr2 determined above.

The results of this analysis are summarized inTable 10-4.7 and compared with the previouslyreported test result.

Comparison of the previously reported test resultwith the more exact value determined by eq.(10-4.13) indicates an error in the previously re-ported test result that is greater than the magnitudeof the previously reported expanded uncertainty.In this case, failure to use the more correct engi-neering relationships would lead to presentationof an uncertainty interval for the test result thatdoes not encompass the true value. This illustrates



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Table 10-4.7 Evaluation of Analysis Error

Symbol Parameter Description Units Average Value

h1 Stagnation enthalpy at measured Btu/lbm 124.27inlet total temperature

h2 Stagnation enthalpy at measured Btu/lbm 231.06exit total temperature

Pr1 Relative pressure at measured . . . 1.2147inlet total temperature

Pr2 Relative pressure at exit . . . 7.8931corresponding to isentropiccompression

h2s Stagnation enthalpy at exit Btu/lbm 212.35corresponding to isentropiccompression

� Computed adiabatic compressor . . . 0.8248efficiency using eq. (10-4.13)

� Previously reported adiabatic . . . 0.8355compressor efficiency usingeq. (10-4.1)

UR Previously reported expanded . . . 0.00772uncertainty in the test result

. . . Difference between results from . . . −0.0107eq. (10-4.13) and eq. (10-4.1)

the importance of applying all known engineeringcorrections and using appropriate engineering rela-tionships as part of the test results analysis process.

10-5 PERIODIC COMPARATIVE TESTING

10-5.1 Problem Definition

Periodic testing of equipment is a common situa-tion where measurement uncertainty must be con-sidered. For this example, a pump is consideredto perform consistently if the supply pressuremeasured at 100% of rated flow is consistent withprior test results. The pump design data is pre-sented in Table 10-5.1-1 and the test data is pre-sented in Table 10-5.1-2 After each test, it is neces-sary to evaluate the test results. As part of thiseffort the effect of measurement uncertainty mustbe considered. For the first test in Table 10-5.1-2the available information is limited since it wasa factory test. (See Figs. 10-5.1-1 and 10-5.1-2.)

The conclusions that can be drawn from Fig.10-5.1-3 are as follows:

(a) The pump is operating consistently whencompared to the factory test results since the uncer-tainty bands overlap.

(b) The pump is operating better than the mini-mum required design condition. The confidence forthis conclusion is better than 95%.

62


The performance of a hydraulic system can beevaluated using Bernoulli’s equation.

H pP�

+v 2

2g+ z

where

Hp total head, mPp system pressure, Pa�p fluid density, kg/m3

vp fluid velocity, m/sgp gravitational constant, 9.81 m/s2

z1p elevation, m

The effective head produced at a specified pres-sure is a measure of pump performance. Theeffective head in terms of the measured test param-eters and physical property data is

�P p P2 +8Q2�

2 d 42

+ (z2 − z1) �g

where

d2p inside pipe diameter, m�Pp pressure change between taps 1 and 2, PaP2p pressure at tap 2, PaQp fluid flow rate, m3/sz2p elevation of pressure tap 2, mz1p elevation of tap 1 or free-surface eleva-

tion, m

The density can be estimated numerically usingthe relationship [6],

� p 766.17 + 1.80396 TK −3.4589 T2

K

1000(10-5.1)

where

TKp TC + 273.15 p the absolute temperatureand TC is the fluid temperature in Celsius

Estimates of the density, �, using eq. (10-5.1)are reported to have a systematic standard uncer-tainty of 0.587 kg/m3. The random error in thecurve fit was judged to be negligible and so therandom uncertainty is set to zero.

During testing it is not feasible to operate exactlyat a specified operating condition. For a pumptest, the applied flow might be slightly differentfor each test. This will result in a slight changein the resultant differential pressure. The variation



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Table 10-5.1-1 Pump Design Data (TC � 20°C)

Flow Differential Pressure

m3/s gpm Percent Rated Flow kPa psi

0.000 0 0 827 119.90.126 2000 100 689 99.90.189 3000 150 552 80.1

Table 10-5.1-2 Summary of Test Results

Field TestsFactory Test

[Note (1)] A B C D E

Raw DataFlow, m3/s 0.126 0.125 0.126 0.130 0.123 0.129P exit, Pa . . . 840000 845000 820000 836000 841000d exit, m . . . 0.254 0.254 0.254 0.254 0.254z exit, m . . . 2.40 2.40 2.40 2.40 2.40z inlet, m . . . 15.00 15.00 15.00 15.00 15.00T, °C 20.15 19.71 20.01 20.31 20.78 21.10

Resultants�, kg/m3 997.7 997.8 997.7 997.7 997.6 997.5�P, Pa 712000 718000 724800 707300 710200 726400�P, psi 103.3 104.1 105.1 102.6 103.0 105.4

NOTE:(1) The factory test data only provides resultant information.

Fig. 10-5.1-1 Installed Arrangement

63




--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Fig. 10-5.1-2 Pump Design Curve With Factory and Field Test Data Shown

Fig. 10-5.1-3 Comparison of Test Results With Independent Control Conditions

in flow may be handled by normalizing the testresults. This is accomplished by adding an addi-tional random term. A normalization coefficientcan be estimated from the factory test data inTable 10-5.1-1. A best fit correlation of the data is

�P p 827,000 − 376,000Q − 5,710,000Q2 (10-5.2)

The normalizing coefficient is the slope of eq.(10.5.2) at the specified test conditions.

64


�PN p b(QN − Q) + �P (10-5.3)

where

�PNp expected pressure change based on thenominal (specified) test conditions, Pa

�Pp measured pressure change, PaQNp nominal (specified) test flow rate, m3/s

Qp measured test flow rate, m3/sbp slope of eq. (10-5.2), Pa · s/m3



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


The slope of eq. (10-5.2) is

∂�PN

∂Q �QN

p b p −376,000 − 11,420,000 QN

At the 100% rated flow condition (see Table 10-5.1-1), QN p 0.126 m3/s, and the value for theslope is b p −1 810 000 Pa · s/m3. The final datareduction equation becomes:

�P p P2 +8Q2�

2d 42

+ (z2 − z1) �g + b(QN − Q)

10-5.2 Comparison With Independent Control

The field test data for the pump can be comparedwith the factory test results or the minimum ratedpressure output. For this type of evaluation, theuncertainties are independent and a simple com-parison of the test results with the benchmarkvalue is adequate. The uncertainty is calculatedusing the method in subsections 7-1 through 7-4.The partial derivatives necessary to estimate thesensitivity coefficients for this problem are

Symbol,Xi Formulas for Absolute Sensitivity

P2∂�P∂P2

p 1

Q16Q�

2d42

− b

d2−32Q2�

2d52

�8Q2

2d42

+ (z2 − z1)g

z1 �g

z2 − �g

b QN − Q

TK 1.80396 −6.9178TK

1000

The temperature sensitivity coefficient is com-puted using the chain rule:

∂�PN

∂TKp

∂�PN

∂�∂�

∂TKp � 8Q2

2d42

+ (z2 − z1) g��1.80396 −

6.9178 TK

1000 �65


The uncertainty for each test can be calculatedas shown in Table 10-5.2-1. Table 10-5.2-2 showsthe nominal value, and the systematic, random,and total uncertainties for �P. The uncertaintiesfor each test are presented in Table 10-5.2-3. Theresults are plotted in Fig. 10-5.2.

10-5.3 Comparative Uncertainties

Correlation of terms is an important consider-ation in comparison testing [23, 24] where twodifferent operating conditions or constructions arebeing compared by use of a ratio

� palt

control(10-5.4)

where�p ratio of two resultants

altp alternate (or variable) resultantcontrolp resultant used for the baseline or

controlFor the pump test C, with test A considered

the control test, eq. (10-5.4) becomes

�A p707 kPa718 kPa

p 0.985

The test results for the comparative analysis aresummarized in Table 10-5.2-3.

The uncertainty for the comparative analysis canbe computed using the method from subsection8-1. The partial derivatives for eq. (10-5.4) are

∂�∂a

p�

a

∂�∂c

p−�c

These may be combined with the partial deriva-tives presented earlier to estimate the sensitivitycoefficients for the comparative analysis.

�P2a

p�

�Pa

∂�Pa

∂P2a

�Qap

��Pa

∂�Pa

∂Qa�d2

a

p�

�Pa

∂�Pa

∂d2a

��ap

��Pa

∂�Pa

∂�a�z2

a

p�

�Pa

∂�Pa

∂z2a

�z1a

p�

�Pa

∂�Pa

∂z1a

�bap

��Pa

∂�Pa

∂ba�TC

a

p�

�Pa

∂�Pa

∂Tca

�p2c

p−��Pc

∂�Pc

∂P2c

�Qcp

−��Pc

∂�Pc

∂Qc�d2

a

p−��Pc

∂�Pc

∂d2c



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Table 10-5.2-1 Uncertainty Propagation for Comparison With Independent Control



(in Parameter Units) (in Result Unit Squared)



Nominal Uncertainty, Uncertainty, Sensitivity, Contribution, Contribution,Symbol Description Units Value bX

isX

i�i (bX

i�i )

2 (sXi�i )

2

Q Flow m3 0.125 3.0�10−3 1.0�10−3 1.86�106 3.11�107 3.46�106

P2 Exit pressure Pa 840.0�103 3500 3000 1.0 1.23�107 9.0�106

d2 Exit diameter m 0.254 1.0�10−3 0 −47810 2.29�103 0z2 Exit elevation m 2.40 0.0125 0 9786 1.50�104 0z1 Inlet elevation m 15.00 0.0125 0 −9786 1.50�104 0Tc Fluid temperature °C 19.71 0.25 0.10 26.76 44.8 7.16� Fluid density kg/m3 997.8 0.6 0 −120.5 5.23�103 0b Correlation Pa·s/m3 −1.82�106 5000 0 1.0�10−3 25.0 0

coefficient

Table 10-5.2-2 Summary: Uncertainties in Absolute Terms

CombinedAbsolute Absolute Standard

Systematic Random Uncertainty TotalStandard Standard of the Absolute

Calculated Uncertainty, Uncertainty, Result, Uncertainty,Symbol Description Units Value bR sR uR UR,95

�P Differential pressure Pa 718,000 6,590 3,530 7,476 14,951

Table 10-5.2-3 Summary of Results for Each Test

�P, bX, sX, U�P,95,Test kPa kPa kPa kPa � b� s� U�,95

Factory Test 712 . . . . . . . . . 0.9916 0.0091 0.0049 0.0207A 718 6.5 4 15 1.0000 0.0048 0.0069 0.0168B 725 6.5 4 15 1.0097 0.0049 0.0070 0.0170C 707 6.5 4 15 0.9847 0.0049 0.0069 0.0169D 710 6.5 4 15 0.9889 0.0048 0.0069 0.0168E 726 6.5 4 15 1.0111 0.0049 0.0070 0.0170

66




--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Fig. 10-5.2 Comparison of Test Results Using the Initial Field Test as the Control

The uncertainty of the comparison ratio, �, whenall of the systematic standard uncertainty termsare correlated and the corresponding sensitivitycoefficients are equal (i.e., �i,alt p �i,control) is zero.For eq. (10-5.4) the systematic standard uncertaintysummation based on eq. (8-1.2).

b2� p ��P2

a

bP2a�2 + ��Qa

bQa�2 + ��d2a

bd2a�2

+ ��ab�a�2 + ��z2

a

bz2a�2 + ��z1

a

bz1a�2

+ ��Tca

bTca�2 + ��BN

a

bBNa�2 + ��P2

c

bP2c�2

+ ��QcbQc�2 + ��d2

c

bd2c�2 + ��c

b�c�2

+ � �z2c

bz1c�2 + ��TC

c

bTCc�2 + ��BN

c

bBNc�2

+ 2�P2a

�P2c

bP2aP2

c

+ 2�Qa�Qc

bQaQc+ 2�d2

a

�d2c

bd2ad2

c

+ 2��a��c

b�a�c+ 2�z2

a

�z2c

bz2az2

c

+ 2�z1a

�z1c

bz1az1

c

+ 2�Tca

�Tcc

bTca

TccTc

c

+ 2�bNa

�bNc

bbNabN

c

67


For this example, all of the systematic standarduncertainties are considered fully correlated exceptfor the exit pressure that is only partially corre-lated. The partial correlation for the pressure mea-surement must be derived from the elementaluncertainties.

For this program the test procedure requiresthat the pressure gage be calibrated just prior toconducting the test. Standard protocol is to bringa replacement gage to the test location on the dayof the test and replace the gage just prior to thetest. Thus, each test is conducted with a differentgage. The absolute systematic standard uncer-tainty, bP, associated with each gage is 3,500 Pa.Since the gage is randomly selected and severalgage brands are used, much of the systematic erroris not correlated. The exception is the calibrationstandard uncertainty, which is known to be 2,500Pa. This calibration standard uncertainty is treatedas a fully correlated standard uncertainty associ-ated with the pressure gage.

For test C, with test A considered the controltest, the absolute random standard uncertaintycomponent, SR, from Table 10-5.3-1 is 0.0069. Theabsolute systematic standard uncertainty of theresult (from Tables 10-5.2-3 and 10-5.3-1) would be



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Table 10-5.3-1 Uncertainty Propagation for Comparative Uncertainty

��p

��P�xi �i bi (bi�i)

2 sXi

(sXi�i)

2

Variable Test C

Q 1.39�10−6 1.86�106 2.590 0.003 6.04�10−5 0.001 6.71�10−6

P2 1.39�10−6 1 1.39�10−6 3,500 2.37�10−5 3,000 1.74�10−5

d2 1.39�10−6 −47,810 −0.0665 0.001 4.42�10−9 0.0 0.0z2 1.39�10−6 9,786 0.0136 0.0125 2.89�10−8 0.0 0.0z1 1.39�10−6 −9,786 −0.0136 0.0125 2.89�10−8 0.0 0.0TC 1.39�10−6 26.76 0.0372 0.25 0.0 0.1 0.0� 1.39�10−6 −120.5 −1.68�10−4 0.6 1.02�10−8 0.0 0.0b 1.39�10−6 0.0010 1.39�10−9 5,000 4.83�10−11 0.0 0.0

Control Test A

Q −1.37�10−6 1.86�106 −2.55 0.003 5.85�10−5 0.001 6.50�10−6

P2 −1.37�10−6 1 −1.37�10−6 3,500 2.30�10−5 3,000 1.69�10−5

d2 −1.37�10−6 −47,810 0.0655 0.001 4.29�10−9 0.0 0.0z2 −1.37�10−6 9,786 −0.0134 0.0125 2.81�10−8 0.0 0.0z1 −1.37�10−6 −9,786 0.0134 0.0125 2.81�10−8 0.0 0.0TC −1.37�10−6 26.76 −3.66�10−5 0.25 0.0 0.1 0.0� −1.37�10−6 −120.5 1.65�10−4 0.6 9.80�10−9 0.0 0.0b −1.37�10−6 0.0010 −1.39�10−9 5,000 4.69�10−11 0.0 0.0

� (RSS value) 0.0129 0.0069

Table 10-5.3-2 Sensitivity Coefficient Estimates for Comparative Analysis

�ia

bia

�ic

bic

2�ia�i

cbi

aic

Q 2.593 0.003 −2.552 0.003 −1.19�10−4

P2 1.39�10−6 2500 −1.37�10−6 2500 −2.38�10−5

d2 −0.0719 0.001 0.0655 0.001 0.0z2 0.0136 0.0125 −0.0134 0.0125 0.0z1 −0.0136 0.0125 0.0134 0.0125 0.0TC 3.78�10−5 0.25 −3.67�10−5 0.25 0.0� −1.68�10−4 0.6 1.65�10−4 0.6 0.0b 5.56�10−9 5000 −1.37�10−9 5000 0.0

� (RSS sum) −1.43�10−4

bR p �(0.0129)2 + (−1.43 � 104) p 4.84 � 10−3

The combined standard uncertainty of the re-sult is

uR p �(4.84 � 10−3)2 + (6.9 � 10−3)2 p 8.43 � 10−3

The total comparative uncertainty

UR p 2uR p 0.0169

68


The uncertainty for each test can be calculatedas shown in Tables 1-5.3-1 and 10-5.3-2. The uncer-tainties for each test are presented in Table 1-5.2-2. The results are plotted in Fig. 10-5.2. The testconclusion that can be drawn from this figure isthe pump is operating consistently when comparedsince the uncertainty bands overlap.



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Section 11References

[1] ISO “Guide to the Expression of Uncertaintyin Measurement.” Geneva: International Or-ganization for Standardization; 1995.

[2] ASME PTC 19.1, Measurement Uncertainty.New York: The American Society of Mechani-cal Engineers; 1985.

[3] Moffat, R. J. “Identifying the True Value —The First Step in Uncertainty Analysis.” ISApaper 88-0729.

[4] Steele, W. G., et al. Use of Previous Experi-ence to Estimate Precision Uncertainty ofSmall Sample Experiments. AIAA Journal, 31:1891–1896; October 1993.

[5] Brownlee, A. K. Statistical Theory and Method-ology in Science and Engineering, 2nd edition.New York: John Wiley and Sons; 1967.

[6] Figliola, R. S., and D. E. Beasley. Theory andDesign for Mechanical Measurements. 3rd edi-tion. New York: John Wiley and Sons; 2000.

[7] Moffat, R. J. “Contributions to the Theory ofa Single-sample Uncertainty Analysis.”Transactions of the ASME: Journal of Fluids En-gineering, 104: 250–260; June 1982. New York:The American Society of Mechanical Engi-neers.

[8] James, M. L. Applied Numerical Methods forDigital Computation, 3rd edition. New York:Harper Collins; 1992.

[9] Coleman, H. W. and W. G. Steele. Experimen-tation and Uncertainty: Experimentation Uncer-tainty Analysis for Engineers, 2nd edition. NewYork: John Wiley & Sons; 1999.

[10] Brown, K. K., et al. Evaluation of CorrelatedBias Approximations in Experimental Uncer-tainty Analysis. AIAA Journal, 34: 1013–1018;May 1996.

[11] Steele, W. G., et al. Asymmetric SystematicUncertainties in the Determination of Experi-mental Uncertainty. AIAA Journal, 34: 1458–1463; July 1996.

69


[12] Wyler, J. S. Estimating the Uncertainty of Spa-tial and Time Average Measurements. Trans-actions of the ASME: Journal of Engineering forPower: 473–476; October 1975.

[13] Holman, J. P. Experimental Methods for Engi-neers, 6th edition. New York: McGraw-Hill;1994.

[14] ISO/TR 7066-1: 1997 (E). “Assessment of Un-certainty in Calibration and Use of Flow Mea-surement Devices — Part 1: Linear Calibra-tion Relationships.” Geneva: InternationalOrganization for Standardization; 1997.

[15] ISO 7066-2: 1988(E). “Assessment of Uncer-tainty in the Calibration and Use of FlowMeasuring Devices — Part 2: Non-linear Cali-bration Relationships.” Geneva: Interna-tional Organization for Standardization;1988.

[16] Montgomery, D. C., and E. A. Peck. Introduc-tion to Linear Regression Analysis. 2nd edition.New York: John Wiley and Sons; 1992.

[17] Price, M. L. “Uncertainty of Derived Resultson X-Y Plots,” ISA paper No. 93-107.

[18] Fuller, W. A. Measurement Error Models. NewYork: John Wiley and Sons; 1987.

[19] ASME PTC 19.1, Test Uncertainty. New York:The American Society of Mechanical Engi-neers; 1998.

[20] NIST Technical Note 1297. Guidelines for Eval-uating and Expressing the Uncertainty of NISTMeasurements. 1994.

[21] Fluid Meters — Their Theory and Application.6th edition. New York: American Society ofMechanical Engineers; 1971.

[22] Chakroun, W., et al. “Bias Error ReductionUsing Ratios to a Baseline Experiment —Heat Transfer Case Study. Journal of Ther-mophysics and Heat Transfer, 7: 754–757; Octo-ber–December 1993.



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[23] Coleman, H. W., W. G. Steele, and R. P. Tay-lor. “Implications of Correlated Bias Uncer-tainties in Single and Comparative Tests.”Transactions of the ASME: Journal of Fluids En-gineering, 117: 552–556; New York: The Amer-ican Society of Mechanical Engineers. Decem-ber 1995.

[24] Hahn, G. “Understanding Statistical Inter-vals.” Industrial Engineering, 45–48; Decem-ber 1970.

[25] Thompson, R. W. “On a Criterion for Rejec-tion of Observations and the Distribution ofthe Ratio of the Deviation to Sample StandardDeviation.” Annals of Mathematical Statistics,6: 214–219; 1935.

[26] Grubbs, F. E. “Procedures for Detecting Out-lying Observations in Samples.” Technome-

70


trics, 11: 1; February 1969.[27] Steele, W. G., et al. “Computer-Assisted Un-

certainty Analysis.” Computer Applications inEngineering Education, 1997.

[28] Steele, W. G., et al. “Comparison of ANSI/ASME and ISO Models for Calculation of Un-certainty.” ISA Transactions, 33: 339–352;1994.

[29] Strike, W. T., and R. H. Dieck. “Rocket Im-pulse Uncertainty.” Proceedings of the 41stISA International Instrumentation Sympo-sium, Denver, 1995.

[30] Coleman, H. W., and W. G. Steele. “Engi-neering Application of Experimental Uncer-tainty Analysis.” AIAA Journal, 33: 1888–1896;October 1995.



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Section 12Bibliography

[1] Benedict, R. P. Fundamentals of Temperature,Pressure, and Flow Measurements. 3rd. edition.New York: John Wiley and Sons, 1984.

[2] ASME MFC-2M, Measurement Uncertainty forFluid Flow in Closed Conduits. New York: TheAmerican Society of Mechanical Engineers;1983.

[3] Hayward, A. T. J. “Repeatability and Accu-racy.” Mechanical Engineering Publication,Ltd., 1977.

[4] Draper, N. R., and H. Smith. Applied Regres-sion Analysis. 2nd. edition. New York: JohnWiley and Sons; 1981.

[5] Williams, E. J. Regression Analysis. New York:John Wiley and Sons; 1959.

[6] Natrella, M. G. “Experimental Statistics.” Na-tional Bureau of Standards Handbook 91, 1963.

[7] Benedict, R. P. “Engineering Analysis of Ex-perimental Data.” Transactions of the ASME:

71


Journal of Engineering for Power. New York:The American Society of Mechanical Engi-neers; January 1969, p. 21.

[8] Berkson, J. “Estimation of Linear Functionfor a Calibration Line.” Technometrics, 11: (4);November 1969.

[9] Mandel, J. “Fitting Straight Lines When BothVariables are Subject to Error.” Journal ofQuality Technology, 15: (1); January 1984.

[10] AGARD Report AG-237. “Guide to In-flightThrust Measurement of Turbojets and FanEngines.”

[11] Davies, O. L. Design and Analysis of IndustrialExperiments. 2nd edition, New York:Hafner, 1967.

[12] Hald, A. Statistical Theory With EngineeringApplications. New York: John Wiley andSons, 1952.



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ASME PTC 19.1-2005

Nonmandatory Appendix AStatistical Considerations

A-1 UNDERSTANDING STATISTICAL INTERVALS

It is often desirable to use the terms confidence,tolerance, and prediction interval. (The use of tolerancein this instance is different from that of PTC 1.)An eloquent treatment of these terms is given byGerald Hahn [24] in his 1970 paper, which isreproduced here by permission.

Statistical intervals are frequently misunderstoodand misused. Here is an explanation of when touse confidence, tolerance, and prediction intervals.

Engineers have come to appreciate that fewthings in life are known exactly. The most theycan do is obtain an estimate and construct aninterval which, with a high probability, containsthe quantity of interest. This article describes threedifferent types of statistical intervals and showswhere each should be used.

The three intervals are: (1) a confidence intervalto contain a population mean, (2) a toleranceinterval to contain a specified proportion of thepopulation, and (3) a prediction interval to containall of a specified number of future observations.

Many nonstatistical users of statistics are wellacquainted with confidence intervals. Some arealso aware of tolerance intervals, but most nonstat-isticians know very little about prediction intervalsdespite their practical importance. A frequent mis-take is to calculate a confidence interval on thepopulation mean when the actual problem callsfor a tolerance interval or a prediction interval.At other times, a tolerance interval is used whena prediction interval is needed.

This confusion is understandable since most textson statistics devote extensive space to confidenceintervals on population parameters, make limitedreference to tolerance intervals, and almost nevertalk about prediction intervals. This is unfortunatebecause tolerance intervals or prediction intervalsare needed as frequently in industrial applicationsas confidence intervals, and given the requiredtabulations, the procedure for constructing them

73


is no more difficult. Table A.1 lists the informationneeded to construct all three intervals.

CONFIDENCE INTERVAL FOR THE POPULATIONMEAN

The sample mean y is an estimate of the un-known mean �, but differs from it because ofsampling fluctuations. However, it is possible toconstruct a statistical interval known as a confi-dence interval for the population mean �. Thisinterval contains � with a specific probability. Thisprobability is known as the associated confidencelevel. Thus a 95 percent confidence interval onthe population mean is an interval which contains� with a probability of 0.95. It is calculated as:

y ± cM (n)s,

where cM (n) is obtained from the first column ofTable A-1.1 as a function of n, the sample size.For the example shown in the box cM (5) p 1.24and the 95 percent confidence interval for � is:

50.10 ± (1.24)(1.31).

Consequently, one can be 95 percent confidentthat the interval 48.48 to 51.72 contains the un-known value of �. More precisely, over a largenumber of samples, the interval calculated in thismanner will contain the unknown mean 95 percentof the time.

TOLERANCE INTERVAL TO CONTAIN A SPECIFICPROPORTION OF THE POPULATION.

Instead of, or in addition to, a confidence intervalto contain �, many applications require an intervalto enclose a specific proportion of the population.For a normal distribution, if � and � are knownexactly, it can be stated that 90 percent of thepopulation is located in the interval.



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Table A-1.1 Factors for Calculating the Two-Sided 95% Probability Intervals for A NormalDistribution

Factors for Confidence Factors for Tolerance IntervalNumber of Interval to Contain to Contain at Least 90%, Factors for Prediction Interval to Contain the

Given the Population Mean 95% and 99% of the Values of All of 1, 2, 5, 10, and 20 FutureObservations � Population Observations

n cM(n) cT,90(n) cT,95(n) cT,99(n) cP,1(n) cP,2(n) cP,5(n) cP,10(n) cP,20(n)

4 1.59 5.37 6.37 8.30 3.56 4.41 5.56 6.41 7.215 1.24 4.28 5.08 6.63 3.04 3.70 4.58 5.23 5.856 1.05 3.71 4.41 5.78 2.78 3.33 4.08 4.63 5.167 0.92 3.37 4.01 5.25 2.62 3.11 3.77 4.26 4.748 0.84 3.14 3.73 4.89 2.51 2.97 3.57 4.02 4.469 0.77 2.97 3.53 4.63 2.43 2.86 3.43 3.85 4.26

10 0.72 2.84 3.38 4.43 2.37 2.79 3.32 3.72 4.1011 0.67 2.74 3.26 4.28 2.33 2.72 3.24 3.62 3.9812 0.64 2.66 3.16 4.15 2.29 2.68 3.17 3.53 3.8915 0.55 2.48 2.95 3.88 2.22 2.57 3.03 3.36 3.6920 0.47 2.31 2.75 3.62 2.14 2.48 2.90 3.21 3.5025 0.41 2.21 2.63 3.46 2.10 2.43 2.83 3.12 3.4030 0.37 2.14 2.55 3.35 2.08 2.39 2.78 3.06 3.3340 0.32 2.05 2.45 3.21 2.05 2.35 2.73 2.99 3.2560 0.26 1.96 2.33 3.07 2.02 2.31 2.67 2.93 3.17� 0 1.64 1.96 2.58 1.96 2.24 2.57 2.80 3.02

A two-sided 95 percent interval is y ± c(n)s, where c(n) is the appropriate tabulated value and y and s are the mean and thestandard deviation of the given sample of size n.

� ± 1.64 �

However, if only sample estimates y and s ofthe population values � and � are given, the bestthat can be stated is that with a chosen probability(say 0.95) the interval contains at least 90, 95, or99 percent of the population. Such an interval iscalled a tolerance interval and can be calculatedfor a normal population with the help of the factorscT,90(n), cT,95(n), and cT,99(n), shown in columns 2,3, and 4 of Table A-1.1.

For example, it can be stated with 95 percentconfidence that the interval:

y ± cT,90 (n)s

contains at least 90 percent of a normal populationThe tolerance interval for the example in the

box may be calculated as:

50.10 ± (4.28)(1.31)

or 44.49 to 55.71 where cT,90(n)s p 4.28. Thus, onemay be 95 percent confident that the precedinginterval contains at least 90 percent of the sampledpopulation.

74

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The fact that both a population proportion (orpercentage) and a statistical probability (also apercentage) are associated with a tolerance intervalis sometimes confusing to the engineer. The firstof these numbers refers to the proportion (orpercentage) of the population that the intervalis to contain. The second number specifies theprobability that the calculated interval really con-tains at least the specified proportion of the popula-tion. When � and � are known exactly, an intervalto contain a specified proportion of the populationmay still be of interest, but, in this case, there isno longer any uncertainty associated with theproportion of the population contained in theinterval.

PREDICTION INTERVAL TO CONTAIN ALL OF ASPECIFIED NUMBER OF FUTURE OBSERVATIONS.

Another type of interval is one that will containall the values of one or more future observations.This is known as a prediction interval. The lastfive columns of Table A-1.1 provide values of thefactor cP,k(n) such that all of k future observationsfrom the same normal population will be locatedin the interval:



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Fig. A-1 How the Lengths of the Statistical Intervals for the Example Compare

y ± cP,k (n)s,

with a probability of 0.95.For example, if two additional readings are taken

from the example in the box, k p 2 and n p 5.From Table A-1.1 the factor cP,2(5) p 3.70. Thustwo future units from the sampled population willbe located in the interval:

50.10 ± (3.70)(1.31)

or 45.25 to 54.95, with a probability of 0.95.The relative lengths of the three intervals ob-

tained in the preceding examples are comparedin Fig. A-1. It is seen that for the given sample of5, the confidence interval to contain the populationmean is appreciably smaller than both the toleranceinterval and the prediction interval. Also a toler-ance interval to include at least 90 percent of thepopulation with a probability of 0.95 is somewhatlarger than a prediction interval to contain bothof two future observations.

Inspection of the tabulations indicates that aconfidence interval on the mean is always smallerthan the other two intervals, but that the relativesizes of the tolerance and prediction intervals de-pend upon the proportion of the population tobe contained in the prediction interval. Also, unlikethe other two intervals, the length of a confidence

75


interval approaches zero as the sample size in-creases (the interval converging to the point �).

HOW TO SELECT THE RIGHT INTERVAL.

The statistician’s job is to develop correct proce-dures for answering relevant questions. The engi-neer must decide upon the relevant questions.Once the questions to be answered have beenclearly stated, it should be easy to decide uponthe correct intervals. The following comments areoffered to serve as a guide to the engineer in thisprocess.

The mean is the most commonly used singlevalue to describe a population. For the normaldistribution, the mean � is one of the two parame-ters which uniquely defines the distribution. It isidentical to the median (50 percent point) andmode (most common value) of the distribution.The population mean is therefore of great interestin characterizing product performance, and is oftenused as a standard by which competing processesare compared. Its use for such comparisons isespecially appropriate when it is reasonable toassume that each of the competing processes hasthe same statistical variability (as measured by theprocess standard deviation) and, therefore, thedifferences between processes can be described



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completely by differences in their means. The as-sumption of equal standard deviations is fre-quently made.

Because of random fluctuations, a sample doesnot provide perfect information about the popula-tion mean �. Thus, a confidence interval is estab-lished which contains the unknown value of �with a specified degree of confidence.

If, instead of characterizing typical process per-formance, you are interested in estimating therange of variation of the underlying populationor of the observations in a future sample, then atolerance interval or a prediction interval is needed.Specifically, a tolerance interval is applicable iflimits are needed that contain most of the sampledpopulation, while a prediction interval would beused to obtain limits to contain all of a smallnumber of future units from the population. Thus,an engineer who is concerned with the perform-ance of a mass-produced item, such as a transistoror a lamp, would generally be interested in atolerance interval to enclose a high proportion ofthe sampled population.

In contrast, a prediction interval to contain allof k future observations may be thought of as theastronaut’s interval. A typical astronaut, who hasbeen assigned to a specific number of flights, isgenerally not very interested in what will happenon the average in the population of all spaceflights, of which his happen to be a random sample(confidence interval on the mean), or even whatwill happen in at least 99 percent of such flights(tolerance interval). His main concern is the worstthat will happen in the one, three, or five flightsin which he will be personally involved. Similarly,a turbine engineer who is bidding on an order ofthree units based upon his past experience on fiveunits of the same type, would use a predictioninterval to obtain specification limits to containthe performance parameter for all three units witha high probability. Prediction intervals are alsorequired by the typical customer who purchasesone or a small number of units of a given productand is concerned with predicting the performanceof the particular units he has purchased (in contrastto the long-run performance of the process fromwhich the sample has been selected).

WHERE TO GET MORE INFORMATION.

Standard books on elementary engineering sta-tistics, Reference 4, give prime space to the conceptof confidence intervals and, in many cases, also

76


discuss tolerance intervals, but make no mention ofprediction intervals except in a regression context.Such intervals, however, are discussed in Refer-ences 1, 2, 3, and 5. Further, Reference 3 providesa comprehensive comparison of statistical intervalsfor a normal population (including more detailedtabulations than are given here) and a discussionof methods for constructing the various intervals.This article also considers additional types of statis-tical intervals such as:

A prediction interval to contain a future samplemean,

A prediction interval to contain a future samplestandard deviation,

A confidence interval for the population stan-dard deviation,

A confidence interval for a population percentile.Finally, a new time-sharing computer program

calculates a wide variety of statistical intervals,including confidence, tolerance, and prediction in-tervals, Reference 6.

THE EXAMPLE PROBLEMThe calculation of the three intervals are illustrated

here by the following numerical example. Assume thatreadings obtained on a normally distributed perform-ance parameter based on a random sample of five unitsare: 51.4, 49.5, 48.7, 49.3, and 51.6. From this informa-tion, the sample mean y and the sample standard devia-tion s are calculated by well-known expressions:

y p �n

ip1yi/n p (51.4 + . . . + 51.6)/5 p 50.10

s p � �n

ip1(yi − y)2

n − 1� 1 ⁄2

p ��(51.4 − 50.10)2 + . . . + (51.6 − 50.10)2�(5 − 1) �

1 ⁄2

p 1.31,

where y1, . . ., yn are the values of n given observa-tions.

REFERENCES

1. Hahn, G. J., “Additional Factors for CalculatingPrediction Intervals for Samples from a NormalDistribution.” Journal of the American StatisticalAssociation, 65, December, 1970.

2. Hahn, G. J., “Factors for Calculating Two-SidedPrediction Intervals for Samples from a NormalDistribution.” Journal of the American StatisticalAssociation, 64, September, 1969.



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3. Hahn, G. J., “Statistical Intervals for a NormalPopulation,” Journal of Quality Technology, Vol-ume 2, Number 3, pages 115–125, July 1970;Volume 2, Number 4, pages 195–206, October1970.

4. Natrella, Mary Gibbons, Experimental Statistics,National Bureau of Standards Handbook 91, USGovernment Printing Office.

5. Nelson, W. B., “Two-Sample Prediction,” Gen-eral Electric Company TIS Report 68-C-404, No-vember 1968. (Available from Distribution Unit,PO Box 43, Building 5, Room 237, Schenectady,New York 12305).

6. “Summary Statistics Package — ONE-SAMS***”Document 003401, General Electric InformationService Department, 7735 Old GeorgetownRoad, Bethesda, Maryland.

A-2 WEIGHTING METHOD

Whenever the value of a test result is determinedby several independent methods, then the testresult and its associated uncertainty may be deter-mined by weighting the means, random uncertain-ties, and systematic uncertainties of the variousmethods. The advantages of utilizing the weightingtechniques described in this section are that theuncertainty associated with a weighted test resultwill usually be less than the uncertainty of theresults determined from each of the independentmethods by themselves. Prior to using theweighting techniques described in this section, themeans and their associated uncertainty intervalsfrom each of the independent methods shouldbe compared as discussed in subsection 8-5. Theweighting techniques described in this sectionshould not be employed if the uncertainty intervalsdo not overlap to a significant degree as this isa possible indication of unaccounted for uncer-tainties.

The weighting methods presented in this sectionare based on the following assumptions:

(a) The various methods used to determine thetest result are independent to the extent that thereis no appreciable correlation between the sources ofuncertainty of the various methods.

(b) The assumptions presented in subsection 1-3are valid for each of the independent methods suchthat the uncertainty model presented in subsection1-3 may be used.

Let Xi represent best estimates of a parameterby N measurement methods. Then X, the weightedmean of the measurements, can be given by

77


X p �N

ip1WiXi

where Wi are the required weighting factors.A weighting principle which is statistically valid

and is based on weighting by variances is applica-ble in this case [5]. This is true since the systematicuncertainty component of a parameter BXi

is as-sumed to equal two times the standard deviation(square root of the variance) of the possible distri-bution of systematic uncertainty. The variance ofthis distribution is (BXi

/2)2. This is combined with(SXi

)2, which is the variance of the average measure-ment. Therefore, UXi

is a combination of variances:

UXip 2�

BXi

2 2

+ (SXi)2�

1 ⁄2

Therefore,

Wi p 1UXi

2

�N

ip1 1UXi

2

(A-1)

where UXirepresents the uncertainties of Xi.

For two measurement methods with the meansX1 and X2, eq. (A-1) yields

W1 p�UX2

�2

�UX1�2 + �UX2

�2

and

W2 p�UX1

�2

�UX1�2 + �UX2

�2p 1 − W1

Using these same weighting factors, the system-atic and random uncertainties of the weightedmean are given by the root-sum-square relations

BX p � �N

ip1�WiBXi

�2�1⁄2

SX p � �N

ip1�WiSXi

�2�1⁄2



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


These values are combined as UX to obtain theweighted uncertainty of X according to eq. (4-3.5)as follows:

UX p 2��BX/2�2 + �SX�2�1 ⁄2

As usual, X, SX, BX, and UX should be reported.

A-3 OUTLIER TREATMENT

A-3.1 General

All measurement systems may produce spuriousdata points. These points may be caused by tempo-rary or intermittent malfunctions of the measure-ment system. Errors of this type should not beincluded as part of the uncertainty of the measure-ment. Such points are considered to be meaninglessas steady-state test data, and should be discarded.Figure A-3.1 shows a spurious data point calledan outlier.

Fig. A-3.1 Outlier Outside the Range of Acceptable Data

78


All data should be inspected for spurious datapoints as a continuing check on the measurementprocess. To ease the burden of scanning largemasses of data, computerized routines are availableto scan steady state data and flag suspected outli-ers. The suspected outliers should then be subjectedto an engineering analysis.

The effect of outliers is to increase the standarddeviation of the system. Tests are available todetermine if a particular point from a sample isan outlier. In most of these tests, the probabilityfor rejecting a good point is set at 5%. This meansthat the odds against rejecting a good point are20 to 1 (or less). The odds could be increased bysetting the probability of rejecting a good datapoint lower. However, this practice decreases theprobability of rejecting bad data points. For smallsamples, bad data points are hard to identify.

Two tests are in common usage for determiningwhether or not spurious data are outliers. Theseare the Thompson � Technique [25] and the GrubbsMethod [26]. The Thompson � Technique is excel-lent for rejecting outliers, but also may reject some



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


good values. The Grubbs Method does not rejectas many outliers but the number of good pointsrejected is smaller. In this Supplement, the Modi-fied Thompson � Technique1 described below isrecommended for identifying suspected outliers.

The suspected outliers should then be subjectedto an engineering evaluation to determine thecause of the outliers. The engineering evaluationshould include an analysis of the instrumentation,the physics of the measurement methods em-ployed, the expected temporal and spatial varia-tion/profiles of the parameters being measured,data from similar tests, etc. If there is valid engi-neering justification, the suspected outliers maybe removed from the analysis of the test resultand its associated uncertainty. The removal ofthese outliers should be documented within thetest report. If there is not valid engineering justifica-tion to remove the suspected outliers and if re-moval of the outliers will significantly changethe test result and its associated uncertainty, thevalidity of the test should be questioned.

A-3.2 Thompson � Technique (Modified)1

Consider a sample (Xi) of N measurements. Thesample standard deviation (SX) and the mean (X)of the sample are calculated.

Suppose Xj, the jth observation, is the suspectedoutlier. Then, the absolute difference of Xj fromthe mean (X) is calculated as

� p |Xj − X |

Using Table A-3.1, a value of � is obtained forthe sample size (N) at the 5% significance level.This limits the probability of rejecting a good pointto 5%. (The probability of not rejecting a bad datapoint is not fixed. It will vary as a function ofsample size.)

The test for the outlier is to compare the differ-ence (�) with the product �SX. If � is larger thanor equal to �SX, we say Xj is an outlier. If � issmaller than �SX, we say Xj is not an outlier.

A-3.3 Example

There were 40 temperature probes installed inone stage of the turbine of a jet engine. The 40

1 Thompson used a different equation for S. The Modified Thomp-son � Technique uses the equation for S as defined in thisSupplement.

79


probe readings were average � and � was calcu-lated from the average for each probe.

26 79 58 24 1 −103 −121 −220−11 −137 120 124 129 −38 25 −60148 −52 −216 12 −56 89 8 −29

−107 20 9 −40 40 2 10 166126 −72 179 41 127 −35 334 −555

334 and −555 are suspected outliers.

To illustrate the calculations for determiningwhether −555 is an outlier, the following steps aretaken.

Mean (X) p 1.125Sample standard deviation (SX) p 140.8Sample size (N) p 40

By the above equation for �,

� p |−55 − 1.125| p 556.125

From Table A-3.1,

�SX p 1.924 � 140.8 p 270.9

Since � > �SX, we conclude that −555 is a possibleoutlier.

Repeating the above procedure for 334,

� p 334 − 1.125 p 332.875

Since � exceeds �SX for the suspected point, 334,we conclude that 334 also is a possible outlier.

This procedure should be repeated for all re-maining data points.

A-4 PARETO DIAGRAMS

A-4.1 General

It is often useful to display the relative sizes ofthe components of a whole with a bar chart. Oneparticular type of bar chart is called a Paretodiagram after Vilfredo Pareto, an Italian economistwho used this type of diagram in his studies ofthe unequal distribution of wealth. Most of theuses today, with extensive activity in the area ofquality control, are attributed to Joseph Juran whodefined the general principle known as the “ParetoPrinciple” — the “Vital Few, Trivial Many.” Mathe-matics were developed that described the distribu-tion, but for the purposes illustrated here, thediagram can be defined as a bar chart with thebars arranged in descending order of size.



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Table A-3.1 Modified Thompson � (At the 5% Significance Level)

N � N �

3 1.1514 1.4255 1.5716 1.6567 1.7118 1.7499 1.777

10 1.79811 1.81512 1.82913 1.84014 1.85015 1.85816 1.86517 1.87118 1.87619 1.88120 1.88521 1.88922 1.89323 1.89624 1.89925 1.90126 1.90427 1.90628 1.90829 1.91030 1.91131 1.91332 1.91533 1.91634 1.91735 1.91936 1.92037 1.92138 1.92239 1.92340 1.92441 1.92542 1.92643 1.92744 1.92745 1.92846 1.92947 1.92948 1.93049 1.93150 1.93151 1.932

52 1.93253 1.93354 1.93455 1.93456 1.93557 1.93558 1.93559 1.93660 1.93661 1.93762 1.93763 1.93764 1.93865 1.93866 1.93867 1.93968 1.93969 1.93970 1.94071 1.94072 1.94073 1.94174 1.94175 1.94176 1.94177 1.94278 1.94279 1.94280 1.94281 1.94282 1.94383 1.94384 1.94385 1.94386 1.94487 1.94488 1.94489 1.94490 1.94491 1.94492 1.94593 1.94594 1.94595 1.94596 1.94597 1.94598 1.94699 1.946

100 1.946� 1.96

80




--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


To apply this to a test uncertainty example, thefirst step is to define the individual systematicand random standard uncertainties of the meanin terms of their relative individual percentagecontributions to the combined standard uncertaintyof a test result, uR. The second step is to createa bar chart which depicts the percentage contribu-tions of individual systematic and random stan-dard uncertainties to the combined standard uncer-tainty in descending order of size.

As elemental sources of standard uncertainty arenot combined as arithmetic sums but are insteadcombined as described in subsection 4-8, derivationof the percentage contribution of an elementalsource of standard uncertainty to the combineduncertainty is computed as the ratio of the squareof the combined standard uncertainty that wouldbe computed if the elemental source were the onlysource of standard uncertainty to the square of thecombined standard uncertainty computed whenaccounting for all sources of uncertainty. Usingthe uncertainty model and associated assumptionspresented in subsection 4-8, expressions for thepercentage contributions from elemental sourcesof standard uncertainty were derived and arepresented below.

The percentage contribution of each elementalsource of random standard uncertainty (sXi

) to thecombined standard uncertainty (uR) is

��isXi�2

u2R

� 100

The percentage contribution of each elementalsource of systematic standard uncertainty (bXi

) tothe combined standard uncertainty is

��ibXi�2

u2R

� 100

The percentage contribution of each correlatedsource of systematic standard uncertainty (bX1X2

),to the combined standard uncertainty is

2�1�2bX1X2

u2R

� 100

A-4.2 Example

The compressor performance example in subsec-tion 10-4 will be used to illustrate the application

81


of Pareto diagrams. The necessary values for pa-rameter sensitivity, systematic standard uncertain-ties, random standard uncertainties, and combinedstandard uncertainty are summarized in the follow-ing table. (Note that these values were obtainedfrom subsection 10-4).

Symbol Value

bP1

0.021bP

20.198

bT1

0.646bT

20.792

bT1T

20.0625

sP1

0.030sP

20.170

sT1

0.300sT

20.600

�P1

−0.0409�P

20.00629

�T1

0.00367�T

2−0.00203

uR 0.00386

For bP1:

bP1% contribution ~ to uR p

��P1bP1

�2

u2R

� 100

p(−0.0409 * 0.021)2

(0.00386)2 � 100

p 5.0%

Similarly:

bP2p 10.4%, bT1

p 37.7%, bT2p 17.3%

For bT1T2:

bT1T2% contribution ~ to uR p

2�T1�T2

bT1T2

u2R

� 100

p �2(0.00367)(−0.00203)(0.0625)

(0.00386)2 � � 100

p −6.3%

for sP1:

sP1% contribution ~ to uR p

��P1sP1

�2

u2R

� 100

p(−0.0409 * 0.030)2

(0.00386)2� 100

p 10.1%



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Similarly:

sP2p 7.7%, sT1

p 8.1%, sT2p 10.0%

Figure A-4.2.1 illustrates the relative contribu-tions to combined standard uncertainty of individ-ual-parameter systematic and random standarduncertainties in terms of a Pareto chart.

However, since the systematic standard uncer-tainties for T1 and T2 are correlated, the relativepercentages for bT1

, bT2, and bT1T2

should be com-bined as shown in Fig. A-4.2-2.

Fig. A-4.2.1 Pareto Chart for Random and Systematic Standard Uncertainties

82


As can be seen from this diagram, the combina-tion of bT1

, bT2, and bT1T2

is the largest contributorto the combined standard uncertainty, uR, andbP1

the smallest. The ideal end result of an analysissuch as this would be to take corrective action,if possible, to reduce the contribution of majorfactors in the combined standard uncertaintythrough changes in methods, instrumentation, orboth. Although this application of Pareto diagramshas been used to determine the relative contribu-tions to combined standard uncertainty of system-atic and random standard uncertainties, themethod can be applied just as easily to the individ-ual estimates of the elemental errors that contributeto systematic and random standard uncertainties.



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Fig. A-4.2.2 Pareto Chart for Random and Some Combined Systematic Standard Uncertainties(Example 10-4)

83




--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

ASME PTC 19.1-2005

Nonmandatory Appendix BUncertainty Analysis Models

B-1 ISO UNCERTAINTY ANALYSIS MODEL

Nonmandatory Appendix B is adapted from“Computer-Assisted Uncertainty Analysis” [27].The uncertainty model given in the ISO Guide to theExpression of Uncertainty in Measurement (1995)[1] ispresented in this paragraph. The uncertainty modelrequires estimates of the uncertainties for each ofthe elemental error sources for each parameter inthe data reduction equation. These estimates arecombined through use of the ISO model to calculatethe band about the experimental result where thetrue result is thought to lie with C% confidence.The estimates of the elemental errors fall into twocategories: systematic standard uncertainties, bik

forthe systematic errors, and random standard uncer-tainties of the mean, si for the random errorswhere i represents each parameter.

The systematic standard uncertainties are relatedto those systematic errors, that remain after allcalibration corrections are made. Systematic uncer-tainties can be estimated through manufacturerinformation, calibrations, and, in most cases,through sound engineering judgment. There willusually be a set, Ki, of elemental systematic stan-dard uncertainties for each parameter, i. As dis-cussed in para. 4-3.2, each elemental estimate,bik

, is taken to be the prediction of the standarddeviation for a particular distribution of possibleerrors for that particular error source. Typically,these error distributions are assumed Gaussian(normally distributed) or rectangular (uniformlydistributed).

For most engineering predictions of systematicuncertainty, the 95% limits of the possible distribu-tion are estimated rather than the standard devia-tion of the distribution. Obtaining the standarduncertainty from the 95% estimate is simply amatter of dividing the estimate by the appropriatedistribution factor, D [i.e., 2.0 for Gaussian, 1.65 p(1.73)(0.95) for rectangular]. The systematic stan-dard uncertainty for parameter i, bi, is determinedfrom the estimates for the Ki elemental errorsources for that parameter as

84


b2i p �

Ki

kp1b2

ik(B-1.1)

The estimate of the random error for a parameteris the random standard uncertainty of the mean,or the estimate of the error associated with repeatedmeasurements of a particular parameter. The ran-dom standard uncertainty of the mean for eachparameter is determined from Ni measurements as

si p1

�Ni� 1Ni − 1 �

Ni

jp1(Xij

− Xi)2�1 ⁄2

(B-1.2)

where

Xi p�Ni

jp1Xij

Ni(B-1.3)

For the case of a single measurement (Ni p 1),previous information must be used to calculatesi [4].

Consider an experimental result that is deter-mined from I measured variables as

R p R�X1,X2, . . .,XI� (B-1.4)

The ISO Guide defines the combined standarduncertainty of the result as

u2R p �

I

ip1(�ibi)2 + 2 �

I−1

ip1�

I

kpi+1�i�kbik (B-1.5)

+ �I

ip1(�isi)2

where

�i p∂R

∂Xi

(B-1.6)

The first two terms on the right side of eq. (B-1.5)represent the systematic standard uncertainty ofthe result bR [eq. (7-4.1) including the correlated



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


terms], and the third term is the random standarduncertainty of the result sR eq. (7-3.1). The covari-ance of the random errors is assumed to be zero.The covariance of the systematic errors, or thecorrelated systematic standard uncertainty, bik, isdetermined by summing the products of the ele-mental systematic standard uncertainties for pa-rameters i and k that arise from the same sourceand are therefore perfectly correlated [10] (seesubsection 8-1).

In order to obtain the overall uncertainty in theresult, UR, at a specified confidence level, the ISOGuide recommends that the combined standarduncertainty of the result be multiplied by a cover-age factor. The coverage factor is the value fromthe t distribution for the required confidence levelcorresponding to the effective degrees of freedomin the result, �R. The values for t are given inTable B-1.

To find �R, the Welch-Satterthwaite formula isadapted as:

�R p

� �I

ip1��ibi�2 + ��isi�2�

2

�I

ip1 ��isi�4

�si

+ �K

i

kp1

��ibik�4

�bik

�(B-1.7)

where �siis either

�sip Ni − 1 (B-1.8)

or the degrees of freedom of the previous informa-tion if si is estimated [4]. The degrees of freedomof the elemental systematic standard uncertainties�bi

k

may be known from previous information orestimated. The ISO Guide recommends the approx-imation

�bik

p 1⁄ 2 ��bik

bik

−2

(B-1.9)

where the quantity in parentheses is an estimateof the relative variability of the estimate of bik

.For instance, if one thought that the estimate ofbik

was reliable to within ±25%, then

�bik

p 1⁄ 2 (0.25)−2 p 8 (B-1.10)

85


Table B-1 Values for Two-Sided ConfidenceInterval Student’s t Distribution [9]

C� 0.900 0.950 0.990 0.995 0.999

1 6.314 12.706 63.657 127.321 636.6192 2.920 4.303 9.925 14.089 31.5983 2.353 3.182 5.841 7.453 12.9244 2.132 2.776 4.604 5.598 8.6105 2.015 2.571 4.032 4.773 6.869

6 1.943 2.447 3.707 4.317 5.9597 1.895 2.365 3.499 4.029 5.4088 1.860 2.306 3.355 3.833 5.0419 1.833 2.262 3.250 3.690 4.781

10 1.812 2.228 3.169 3.581 4.587

11 1.796 2.201 3.106 3.497 4.43712 1.782 2.179 3.055 3.428 4.31813 1.771 2.160 3.012 3.372 4.22114 1.761 2.145 2.977 3.326 4.14015 1.753 2.131 2.947 3.286 4.073

16 1.746 2.120 2.921 3.252 4.01517 1.740 2.110 2.898 3.223 3.96518 1.734 2.101 2.878 3.197 3.92219 1.729 2.093 2.861 3.174 3.88320 1.725 2.086 2.845 3.153 3.850

21 1.721 2.080 2.831 3.135 3.81922 1.717 2.074 2.819 3.119 3.79223 1.714 2.069 2.807 3.104 3.76824 1.711 2.064 2.797 3.090 3.74525 1.708 2.060 2.787 3.078 3.725

26 1.706 2.056 2.779 3.067 3.70727 1.703 2.052 2.771 3.057 3.69028 1.701 2.048 2.763 3.047 3.67429 1.699 2.045 2.756 3.038 3.65930 1.697 2.042 2.750 3.030 3.646

40 1.684 2.021 2.704 2.971 3.55160 1.671 2.000 2.660 2.915 3.460

120 1.658 1.980 2.617 2.860 3.373� 1.645 1.960 2.576 2.807 3.291

GENERAL NOTES:(a) See [9].(b) Given are the values of t for a confidence level C and number

of degrees of freedom �.

With �R known, the proper t value is obtainedfrom Table B-1 for C% confidence and multipliedby uR from eq. (B-1.5) to obtain the overall uncer-tainty in the result, UR, at a C% confidence level

UR,C p tC � �I

ip1(�ibi)2 (B-1.11)

+ 2 �I−1

ip1�

I

kpi+1�i�kbik + �

I

ip1(�isi)2�

1 ⁄2



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


B-2 LARGE SAMPLE UNCERTAINTY ANALYSISAPPROXIMATION

The method described in subsection B-1 is thestrict ISO method. It has been shown [28-30] thatfor most engineering applications, when the de-grees of freedom for the result from eq. (B-1.7) is9 or greater, t95 can be taken as 2 to a goodapproximation (93% to 95% coverage). Therefore,for large degrees of freedom in the result

UR,95 p 2 � �I

ip1(�ibi)2 (B-2.1)

+ 2 �I−1

ip1�

I

kpi+1�i�kbik + �

I

ip1(�isi)2�

1 ⁄2

86


The first two terms in the brackets in eq. (B-2.1)are the systematic standard uncertainty of theresult bR [eq. (7-4.1) including the correlated sys-tematic standard uncertainty terms], and the thirdterm in the brackets is the random standard uncer-tainty of the result sR eq. (7-3.1). The large sampleuncertainty expression given in eq. (7-5.1) is thenobtained as (including the correlated systematicuncertainty terms)

UR,95 p 2�b2R + s2

R�1⁄2 (B-2.2)



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---

ASME PTC 19.1-2005

Nonmandatory Appendix CPropagation of Uncertainty Through Taylor Series

C-1 INTRODUCTION

Experimental results are not always directly mea-sured. It is quite common for an experimentalresult, r(X1, X2, . . ., Xn), to be defined as a functionof certain variables, X1, X2, . . ., Xn, that are directlymeasured. The aim of this Appendix is to providea method by which the variance of an experimentalresult that is not directly measured, r(X1, X2, . . .,Xn), can be expressed in terms of the variancesand covariances of its arguments, X1, X2, . . ., Xn,which are directly measured.

The approach will be to relate the deviationsin r(X1, X2, . . ., Xn) to deviations in (X1, X2, . . .,Xn) by means of a first order approximation tothe Taylor series expansion for r(x1, x2, . . ., xn)in the neighborhood of the point (�X1, �X2, . . .,�Xn), where �Xi is the true value of the measuredvariable Xi. In order to facilitate this project, thefunction r(x1, x2, . . ., xn) will be assumed to becontinuous with continuous partial derivatives inthe neighborhood of the point (�X1, �X2, . . ., �Xn).

C-2 DEFINITIONS

The primary goal of Nonmandatory AppendixC is to present an expression for the variance inr in terms of the variances and covariances of X1,X2, . . ., Xn. The definitions below for “mean,”“variance,” and “covariance” will be used through-out this Appendix.

The expected value of a function f(X) of arandom variable X is given by

E[f(X)] p ��

-�f(x) p(x) dx

where p(x) is the probability density function for X.The mean (or expected value) �X of the random

variable X presents the special case where f(X) pX, i.e.,

87


E[X] p ��

-�x p(x) dx p �X

The variance �2X of the random variable X pres-

ents the special case where f(X) p (X − �X)2, i.e.,

E[(X − �X)2] p ��

-�(x − �X)2 p(x) dx p �X

2

The expected value of a function f(X1, X2, . . .,Xn) of the random variables X1, X2, . . ., Xn isgiven by

E[f(X1, X2 , . . ., Xn)] p ��

-��

-�. . .

��

-�f(x1 , x2, . . ., xn)

p(x1 , x2 , . . ., xn) dx1 dx2 . . . dxn

where p(x1, x2, . . ., xn) is the joint probabilitydensity function for X1, X2, . . ., Xn.

The covariance �X1X2 of the random variablesX1, X2 presents the special case where f(X1, X2) p(X1 − �X1)(X2 − �X2), i.e.,

E[f(X1, X2)] p ��

-��

-�(x1 − �X1)

(x2 − �X2) p(x1 , x2) dx1 dx2 p �X1X2

C-3 PRELIMINARY CONSIDERATIONS

If any random variable, Y, can be expressed asa linear combination of random variables, Xi, thenthe mean and variance of Y can be expressed interms of the means, variances, and covariances ofthe variables Xi.

Suppose

Y p a0 + a1X1 + a2X2 + . . . + anXn



--`,,``,````````,`,,```,`,,-`-`,,`,,`,`,,`---


Then

E[Y] p E[a0 + a1X1 + a2X2 + . . . + anXn]

or, using the definition of the mean value of arandom variable,

�Y p a0 + a1�X1 + a2�X2 + . . . + an�Xn

The variance of Y will be given as follows:

�2Y p E[(Y − �Y)2]

p E{[a1 (X1 − �X1) + a2 (X2 − �X2)

+ . . . + an (Xn − �Xn)2]}

For example, if n p 3, then

�2Y p E{[a1 (X1 − �X1)

+ a2 (X2 − �X2) + a3 (X3 − �X3)2]}

�2Y p E[a2

1 (X1 − �X1)2 + a22 (X2 − �X2)2

+ a23 (X3 − �X3)2

+ 2a1a2 (X1 − �X1)(X2 − �X2)

+ 2a1a3 (X1 − �X1)(X3 − �X3)

+ 2a2a3 (X2 − �X2)(X3 − �X3)]

or

�2Y p a1

2 �X12 + a2

2 �X22 + a3

2 �X32

+ 2a1a2�X1X2 + 2a1a3�X1X3 + 2a2a3�X2X3

or more generally, the variance of Y will begiven by

�2Y p �

n

ip1 �ai2 �Xi

2 + 2 �n

jpi+1aiaj �XiXj�

C-4 PROPAGATION OF UNCERTAINTY/ERRORTHROUGH TAYLOR SERIES

In subsection C-3 we found that if a randomvariable, Y, could be expressed as a linear combina-tion of the random variables, Xi, the mean andvariance of Y can be expressed in terms of themeans, variances, and covariances of the variablesXi. Now suppose an experimental result, r, isdefined as a function of certain measured variables,X1, X2, . . ., Xn. If the function r(X1, X2, . . ., Xn)can be expressed as a linear combination of themeasured variables X1, X2, . . ., Xn, by means ofa Taylor series approximation to r(X1, X2, . . ., Xn)in the neighborhood of �X1, �X2, . . ., �Xn, then

88


using the results in para. C-3 we will be able toexpress the mean and variance of r(X1, X2, . . ., Xn)in terms of the means, variances, and covariances ofthe variables X1, X2, . . ., Xn.

C-4.1 The First Order Approximation

Suppose r p r(x1, x2, . . ., xn). If we expand rthrough a Taylor series in the neighborhood of�X1, �X2, . . ., �Xn we get

r(x1 , x2 , . . ., xn) p r(�X1, �X2, . . ., �Xn) + �X1 (x1

− �X1) + �X2 (x2 − �X2) + . . . + �Xn (xn − �Xn)

+ higher order terms

where �xi are the sensitivity coefficients given by

�Xi p∂r∂xi

Now, suppose that the arguments of the functionr(X1, X2, . . ., Xn) are the random variables X1, X2,. . ., Xn. Furthermore, assume that the higher orderterms in the Taylor series expansion for r arenegligible compared to the first order terms. Thenin the neighborhood of �X1, �X2, . . ., �Xn we have

r(X1 , X2 , . . ., Xn) ≈ r(�X1, �X2, . . ., �Xn) + �X1 (X1

− �X1) + �X2 (X2 − �X2) + . . . + �Xn (Xn − �Xn)

Consequently

�r ≈ r(�X1, �X2, . . ., �Xn)

and

�2r ≈ �

n

ip1 ��Xi2 �Xi

2 + 2 �n

jpi+1�Xi�Xj �XiXj�

For the special case where the random variablesX1, X2, . . ., Xn are all independent we get

�2r ≈ �

n

ip1�Xi

2�Xi2

Table C-4 presents some useful formulas forpropagating variance through the first order ap-proximation to the Taylor series for an experimen-tal result r.



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Table C-4 Taylor Series Variance Propagation Formulas

Variance (in Absolute Units) Variance (Dimensionless)Function and Absolute Sensitivities and Relative Sensitivities

r p f(x,y)S2

r p �∂r∂x

Sx�2

+ �∂r∂y

Sy�2

V2r p �∂r/r

∂x/xVx�

2

+ �∂r/r∂y/y

Vy�2

�x p∂r∂x

; �y p∂r∂y �x′ p

∂r/r∂x/x

; �y′ p∂r/r∂y/y

r p Ax + By S2r p A2S2

x + B2S2y

V2r p

A2x2V2x + B2y2V2

y

(Ax + By)2�x p A; �y p B

�x′ pAx

Ax + By; �y′ p

ByAx + By

r p1y Sr

2 pS2

y

y4 V2r p V2

y

�y p −1

y2 �y′ p −1

r px

x + y S2r p � ySx

(x + y)2�2

+ � xSy

(x + y)2�2

V2r p

y2(V2x + V2

y)

(x + y)2

�x py

(x + y)2; �y p −

x

(x + y)2 �x′ py

x + y; �y′ p −

yx + y

r px

1 + x S2r p

S2x

(1 + x)4 V2r p

V2x

(1 + x)2

�x p1

(1 + x)2 �x′ p1

1 + x

r p xy S2r p (ySx)2 + (xSy)2

V2r p V2

x + V2y

�x p y; �y p x�x′ p 1; �y′ p 1

r p x2S2

r p 4x2S2x V2

r p 4V2x

�x p 2x�x′ p 2

r p x1⁄2

S2r p

S2x

4x V2r p

V2x

4

�x′ p 1⁄ 2�x p1

2x1⁄2

r p ln xS2

r pS2

x

x2 V2r p � Vx

ln x�2

�x p1x �x′ p

1ln x

r p kxaybS2

r p (akybxa−1Sx)2 + (bkxayb−1Sy)2 V2r p (aVx)2 + (bVy)2

�x p akybxa−1; �y p bkxayb−1 �x′ p a, �y′ p b

GENERAL NOTE: Vx pSx

xVy p

Sy

yVr p

Sr

r

89




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C-4.2 Assessing the Validity of the First OrderApproximation

In para. C-4.1 we assumed that the Taylor seriesexpansion for r could be reasonably approximatedthrough the first order terms. In this paragraphwe will briefly assess the conditions under whichthis approximation is meaningful.

Let’s first consider the simplest case wherer(x). Then

r(x) p r(�x) + �x (x − �x)

+ (1/2!) �xx (x − �x)2 + higher order terms

where

�xx p∂2r

∂x2

In this case the ratio of the second order termin the series to the first order term in the seriesis given by

R p�xx (x − �x)

�x

So that for this case the assumption that R �1reduces to the condition that �xx (x − �x) ��x.

More generally, if the second order terms in theTaylor series expansion are retained, the Taylorseries expansion for r in the neighborhood of �x1,�x2, . . ., �xn becomes

r(x1 , x2 , . . ., xn) p r(�x1, �x2, �xn)

+ �n

ip1�xi (xi − �xi)

+ (1⁄ 2!) �n

jp1�n

kp1�xjxk

(xj − �xj)(xk − �xk)

+ higher order terms

where

�xjxk p∂2r

∂xj∂xk

The second order terms in this expansion maybe compared directly to the first order terms inorder to assess their significance.

90


C-4.3 The Limitation of the Present Approach

If the higher order terms in the Taylor seriesexpansion are not small compared to the firstorder terms, then there is no way to express thevariance in r(X1, X2, . . ., Xn) directly in terms ofthe variances and covariances for X1, X2, . . ., Xnfor an arbitrary joint probability density functionp(x1, x2, . . ., xn).

C-5 PROPAGATION OF SYSTEMATIC ANDRANDOM COMPONENTS OF UNCERTAINTY

The total variance associated with a measuredvariable, Xi, can be expressed as a combinationof the variance associated with a fixed componentand the variance associated with a random compo-nent of the total error in the measurement. In thisparagraph we will relate these two sources ofvariance in the measured variables, Xj, to thevariance in an experimental result r(X1, X2, . . ., Xn).

In subsection 4-2, the total error in a measure-ment was given by

� p +

so that

E[�] p E[ + ]

�� p �

also

E[(� − ��)2] p E{[( − �) + 2]}

�2� p �2

+ �2 + 2�

Assuming the systematic error to be independentof the random component of error this becomes

�2� p �2

+ �2

Now, if we define the random variable Xi asfollows:

Xi p (�i − �i) + �Xi

then

E[Xi] p �Xi

E[(Xi − �Xi)2] p ��i2 p �i

2 + �i2 p �Xi

2



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Assuming fixed errors to be independent ofrandom errors, it can also be shown that

E[(Xi − �Xi)(Xj − �Xj)] p �XiXj p �ij + �ij

These results can be combined with those inpara. C-3.1 to yield

�r2 ≈ �

n

ip1 ��Xi2�i

2 + 2 �n

jpi+1�Xi�Xj �ij�

+ �n

ip1 ��Xi2�i

2 + 2 �n

jpi+1�Xi�Xj �ij�

91


For no correlation among the random errors,with the use of sample statistics, and for a 95%confidence level, this equation leads to eq. (B-1.3).

C-6 THE PROBABILITY DENSITY FUNCTION OF ARESULT

The preceding paragraphs make no assumptionsabout the joint probability density function of themeasured variables Xi. Assuming that the firstorder approximation to the Taylor series expansionfor r is adequate and that the measured variables Xiare jointly normally distributed, the experimentalresult r will also be normally distributed withmean �r and variance �2

r .



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ASME PTC 19.1-2005

Nonmandatory Appendix DThe Central Limit Theorem

Paragraph G.2.1 in [1] states

“The Central Limit Theorem: Suppose Y p

c1X1 + c2X2 + . . . + cNXN p ∑N

ip1ciXi. Then the dis-

tribution of Y will be approximately normal with

expectation E(Y) p ∑N

ip1ciE(Xi) and variance

� 2(X) p ∑N

ip1ci

2� 2 (Xi), where E(Xi) is the expecta-

tion of Xi and � 2(Xi) is the variance of Xi, providedthat Xi are independent and � 2 (Y) is much largerthan any single component ci

2� 2 (Xi) from a non-normally distributed Xi.”

Although not mathematically precise, this is areasonable statement of the central limit theoremfor application purposes. It is a more generalversion of the central limit theorem than whatone would find in elementary statistics textbooks.

92




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D04505

ASME PTC 19.1-2005



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asme 19.1 test uncertainty_05.pdf

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