Theory of Measurements andErrors

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Errors

Florence A. GaleonAssistant Professor

U.P. College of Engineering

Outline

I. IntroductionII. Error and Mistake/BlunderIII. Sources of ErrorIV. Kinds of ErrorIV. Kinds of ErrorV. Accuracy and PrecisionVI. Theory of ProbabilityVII. Most Probable ValueVIII. Illustrative Problems

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Introduction

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Angles Measured by Each Instrument

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800101

8001

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Error and Mistake/Blunder

Error is defined as the difference between the true or established value and the measured value.

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Mistake/Blunder results from misunderstanding the problem, poor judgment, confusion on the part of the observer, or careless procedures, and are indications of incompetence.

Sources of Error

1. Personal Errors these arise because of the limitations of the human senses of sight, touch or hearing.

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2. Instrumental Errors these are due to imperfections in the instruments or accessories with which measurements are taken.

3. Natural Errors these are caused by variations in the phenomena of nature.

Refraction illustration

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Wind effect on instruments

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Magnetic Declination illustration

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Kinds of Errors

Systematic Errors

Systematic errors always follow some definite mathematical or physical law.

They can be computed and their effects reduced

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They can be computed and their effects reduced or eliminated by applying corrections.

Accidental Errors

These are the errors which remain after mistakes and systematic errors have been eliminated.

Accidental ErrorsThey are caused by factors beyond the control of the observer, obey the laws of probability, and are sometimes called random errors.

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There is no absolute way to compute accidental errors or to correct and eliminate them.

Accidental errors are also called compensating errors and are those which are as likely to be positive as they are to be negative, that is, they tend to balance or to compensate one another in a series of measurements.

Terms related to the magnitude of error:

Discrepancy

It is the difference between two measured values of the same quantity.

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Accuracy

It is the degree of conformity with a standard (the "truth").

It relates to the quality of a result.

Precision

It is the degree of refinement in the performance of an operation, or the degree of perfection in the instruments and methods used to obtain a result.

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and methods used to obtain a result.

It is an indication of the uniformity or reproducibility of a result.

It relates to the quality of an operation by which a result is obtained.

Accuracy Versus Precision

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Accuracy Accuracy

Versus Versus

PrecisionPrecision

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Accuracy and Precision: numerical example

For Example: Target Distance Value = 100.000 meters

Values Accurate? Precise?

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100.5 no no100.0 yes no100.555 no yes100.001 yes yes

Accuracy and Precision: numerical example

For Example: Target Angle Value = 1000000

Values Accurate? Precise?

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10030 no no10000 yes no1003030 no yes1000001 yes yes

Theory of Probability

It is based on the following assumptions relative to the occurrences of errors:

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1. Small errors occur more often than large ones and that they are more probable.

Theory of Probability

2. Large errors happen infrequently and are therefore less probable; for normally distributed errors, unusually large ones may be mistakes

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errors, unusually large ones may be mistakes rather than accidental errors.

Theory of Probability

3. Positive and negative errors of the same size happen with equal frequency; that is, they are equally probable.

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4. The mean of an infinite number of observations is the most probable value.

Most Probable Value

Definition:

It refers to a quantity which, based on

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It refers to a quantity which, based on available data, has more chances of being correct than has any other.

Illustrative Problems

1.A UP Engineering professor sent out six groups of GE 10 students to measure a distance between two points marked on the ground. The students came up

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points marked on the ground. The students came up with the following six different values: 250.25, 250.15, 249.90, 251.04, 250.50, and 251.22 meters. Assuming these values are equally reliable and that variations result from accidental errors, determine the most probable value of the distance measured.

Illustrative Problem

2. The observed interior angles of a triangle are:A = 351437,B = 963009, and

A

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B = 963009, and C = 481505. Determine the discrepancy for the given observation and the most probable value of each angle.

B C

Illustrative Problem

3.The angles about a point Q have the following observed values: 1301520, 1423730, and

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1301520, 1423730, and 870740. Determine the most probable value of each angle.

Q

870740

Illustrative Problem

4. Measurement of three horizontal angles about a point are: APB = 123150, BPC = 372920, andCPD = 473630. If the

A

B

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CPD = 473630. If the measurement of the single angle APD is 973700, determine the most probable values of the angles.

C

D

P

Basic Statistics

Definition of Terms:

1. Probable Error

a mathematical quantity giving an indication of precision

2.Residuals or Deviations

The difference between each of the individual measurements from the mean valuemeasurements from the mean value

v = (x-)where v = residual or deviation

x = an observed value = mean of the observed values

3.Standard Error or Standard Deviation

A measure of the amount of variation in the data

= v / (n-1)Where = standard deviation

v = sum of the squares of the residualsn = number of observations

4.Probable Error of a Single Observation

Indicates the degree of precision which may be expected in any single observation made under expected in any single observation made under the same conditions

Es = 0.6745()Where Es = probable error of a single observation

= standard deviation

5.Probable Error of the Mean

Em = Es/ n Em = Es/ nWhere Em = probable error of the mean

Es = probable error of a single observationn = number of observations

6.Error Ratio or Precision of Measurement

The ratio of the error in a measurement to the whole measurementThe ratio of the error to the most probable value The ratio of the error to the most probable value

RP or ER = e/MV or Em/MPVWhere e = error in a measurement

MV = value of the whole measurementEm = probable error of the mean

MPV = most probable value

Precision Building Subsidence Monitoring Survey

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YKT Building in Binondo

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The 6th Monitoring Survey

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