# mece301 01 calibration uncertainty analysis detailednotes (1)

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• Instrument Calibration andUncertainty Analysis

1 Calibration of an Instrument

The static performance charactertistics of an instrument can be determinedby ANSI/ISA Standard 51.1 Process Instrumentation Terminology. Thisstandard gives the definition for the following important characteristics:

1. Accuracy the maximum difference between measurements and a stan-dard (or reference) as determined by testing over the range of the in-strument.

2. Repeatability the maximum difference between repeated measure-ments of the same standard in the same direction of measurement (i.e.increasing or decreasing the input).

3. Resolution the smallest increment you can read on the scale. If youcan see a half increment (i.e. the space between the tick marks), thenthe resolution will be one half of the unit spacing.

4. Range the maximum and minimum values you can measure (for ex-ample: 0 10 mm).

5. Span the algebraic difference between the upper and lower rangevalues.

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• Example 1You wish to determine the accuracy and repeatability of your bathroom

scale in terms of kg. You measure calibration-standard masses (with anaccuracy rating of 0.0005 kg) over the range of the scale. The range of thescale is 0 100 kg. Your scale has 1 kg increments but you can make outhalf increments (i.e. the resolution is 0.5 kg). Following ANSI/ISA 51.1, youfind the results shown in Table 1.

Table 1: Weigh scale calibration data.Scale Reading (kg)

Standard (kg) Up Down Up Down Up Down0.000 0.0 0.0 0.020.000 19.5 21.0 22.0 19.0 19.5 20.040.000 43.0 39.0 39.5 40.0 41.0 39.060.000 60.0 59.5 61.0 58.0 59.5 61.080.000 80.0 82.0 79.5 79.0 80.0 79.5100.000 100.0 99.5 100.0

SolutionThe easiest way to find the accuracy and repeatability is to make a deviationtable shown in Table 2.

Table 2: Weigh scale deviation data.Deviation from standard (kg) Max. Dev. Max. Dev.

Standard (kg) Up Down Up Down Up Down of Ups (kg) of Downs (kg)0.000 0.0 0.0 0.0 0.020.000 -0.5 1.0 2.0 -1.0 -0.5 0.0 2.5 2.040.000 3.0 -1.0 -0.5 0.0 1.0 -1.0 3.5 1.060.000 0.0 -0.5 1.0 -2.0 -0.5 1.0 1.5 3.080.000 0.0 2.0 -0.5 -1.0 0.0 -0.5 0.5 3.0100.000 0.0 -0.5 0.0 0.5

According to ANSI/ISA 51.1 the measured accuracy of the scale is +3.0/-2.0 kg.Notice that the accuracy is not symmetric. If the error was random andnormally distributed, and enough measurements were taken, it would be ex-pected that the accuracy would be symmetric. In this course, we will report

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• the accuracy symmetrically using the most conservative estimate. This wouldbe similar to the accuracy rating given by a manufacturer. This gives an ac-curacy of 3.0 kg or 3% Full Scale (or FS) or Span. We will state theaccuracy this way because: 1) it makes further uncertainty analysis easier,and 2) we rarely have time to repeat the experiment enough times to ensurerandom occurrences do not skew our results (typically a minimum of five upand five down traverses are used).

The repeatability is simply the maximum deviation between repeatedmeasurements of the same value approached from the same direction. In thiscase the repeatability is 3.5 kg or 3.5% of span.

Note: According to ANSI/ISA 51.1, if the accuracy rating of the standardis less than one tenth of the accuracy of the instrument tested, then theaccuracy rating (or inaccuracy) of the standard may be ignored. Since theaccuracy rating of the masses is 0.0005 kg and the accuracy of the scale is3.0 kg, the uncertainty in the masses is simply ignored.

2 Uncertainty in a Measurement

Lets say you know the accuracy of an instrument and you use the instrumentto measure some property. What is the uncertainty in that measurement?This section will explain how to determine the uncertainty in an experimentalmeasurement, but first we must cover a few important definitions.

The error is defined as the difference between the true value and themeasured value. Since we can never know the true value of a property wecan NEVER know the error in the measurement. What we can do is estimatethe uncertainty in our estimate of the true value. The uncertainity is simplythe estimate of the error with a given confidence interval.

The International Standards Organization (ISO) defines two types of un-certainty:

1. Type A Uncertainty (Px) The uncertainty estimated from the data,sometimes this is called precision or repeatability uncertainty. (How-ever, as we will see later, this is not the same repeatability as definedabove in the ANSI/ISA standard. Therefore, I will try to stick to theterm precision when discussing Type A uncertainty.)

2. Type B Uncertainty (Bx) Uncertainties that cant be analyzed from

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• the data and must be estimated from other sources. Sometimes this iscalled systematic or bias uncertainty.

The total uncertainty (Ux) in a measurement is:

Ux =

B2x + P2x (1)

2.1 Precision Uncertainty (Px)

By taking repeated measurements of the property we are trying to determinewe will obtain a distribution of measurements. An important result fromstatistics is the central limit theorem (see Sec. 3.6 in Beckwith). Neglectingany bias error and for a large sample size, the central limit theorem states:

x zc/2 Sxn< < x+ zc/2

Sxn

(2)

where is the true mean value of the property, x is the mean of the mea-surements, zc/2 is the z-score of the normal distribution with c% confidence,Sx is the standard deviation of the measurements, and n is the number of

measurements taken. Therefore, the estimate of the true value is x zc/2 Sxn

with c% confidence. The term zc/2Sxn

is simply the precision uncertainty

(i.e. Px = zc/2Sxn

).

If the sample size is small (n

• Figure 1: Students t-Distribution (Beckwith et al., 2006)

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• Sec. 3.9 in Beckwith). A very common estimate of the bias uncertainty isby using the manufacturers specification for accuracy or the accuracy deter-mined by your own calibration of an instrument. Regardless of the sourceof this estimate, it is important that the estimate of the bias uncertaintyhas the same confidence interval as the precision uncertainty. If we use theANSI/ISA 51.1 definition for accuracy we have no idea what the confidenceinterval is. However, most people assume that the accuracy rating has a95% confidence. In fact this will be a conservative estimate considering thataccuracy is defined as the maximum deviation from the standard.

Other bias errors might not be covered in the accuracy of the instrumentand must be accounted for elsewhere, such as spatial variation of the propertyor any effect the instrument might have on the system.

Example 2You weigh yourself 5 times on the bathroom scale discussed in the previ-

ous example (accuracy of 3.0 kg). Your measurements are: 62.5 kg, 63.0 kg,61.0 kg, 62.0 kg, and 62.0 kg. Estimate your mass and give the uncertaintywith 95% confidence.

SolutionSince the sample size is small the precision uncertainty can be found from

Px = t2,Sxn. (4)

The desired confidence interval is 95%, so = 1 0.95 = 0.05 and

2= 0.025. The number of degrees of freedom are = n 1 = 4. Using

these values, t0.025,4 = 2.776 is found from Figure 1.The standard deviation, Sx, is found from

Sx =

(xi x)2n 1 . (5)

Using the data we get x = 62.1 kg and Sx=0.742 kg.

Therefore, Px = 2.776(0.742 kg)

5= 0.92 kg.

In this case we estimate the bias uncertainty as 3 kg. Therefore,

Ux =B2x + P

2x =

(3 kg)2 + (0.92 kg)2 = 3.1 kg (6)

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• Therefore, your weight is estimated to be 62.1 3.1 kg with a confidenceof 95%.

NOTE: In this case the precision uncertainty had little effect on the totaluncertainty. Therefore, less measurements could have been made withoutmuch compromise in the total uncertainty in the measurement. To make adrastic improvement in the measurement uncertainty a more accurate scalewould be required.

2.3 Single-Sample Precision Uncertainty

Ideally several measurements are taken of a property to minimize the pre-cision uncertainty. However, it may occur that only one measurement ispossible. If this is the case Eq. 3 will no longer hold because Sx and t

2, are

undefined. In this case it is common to estimate the precision uncertaintyas,

Px zc/2e 2e, (7)where e is the standard deviation of measurements that were taken at someother time with the same instrument (e.g. when the instrument was cali-brated), and where the zc/2 2 for a 95% confidence interval.

Example 3You weigh yourself once with the bathroom scale described in Example 1

and obtain a measurement of 62.5 kg. Estimate your mass and give theuncertainty with 95% confidence.

Solution

Since you have only made one measurement you must estimate the pre-cision uncertainty using the single-sample assumption. In this case you canuse the calibration data in Example 1 to find e. The standard deviation ofall the data points in Table 2 is 1.02 kg. Therfore,

Px 2(1.02 kg) = 2.04 kg (8)

andUx =

B2x + P

2x =

(3 kg)2 + (2.04 kg)2 = 3.6 kg (9)

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• Therefore, your weight is estimated to be 62.5 3.6 kg with a confidenceof 95%.

3 Propagation of Uncertainty

If we know the equations that describe a system, it is possible to estimatethe probable uncertainty in a variable from the known uncertainties in othervariables. This is sometimes called an external estimate of error, because itcan be made without doing any experiments.

3.1 Analysis of Equations for Small Errors

Consider an output y that is a function of inputs x1, x2, etc. y = y(x1, x2, x3, . . . , xn)Then, take the derivative

dy =y

x1dx1 +

y

x2dx2 + + y

xndxn (10)

Va