1. Introduction to Instrumentation Engineering
Chapter 1: Measurement Error Analysis
By Sintayehu Challa

2. Goals of this Chapter Differentiate the types of error
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Goals of this Chapter
Differentiate the types of error
Every measurement involves an error
Give an overview of data analysis techniques in instrumentation systems
Understand basic mathematical tools required for this purpose
Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

3. Overview Measurement error analysis Types of errors and uncertainty
Statistical analysis
Gaussian and Binomial distributions
Method of Least Squares
Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

4. Measurement Error Types of errors: Systematic and random errors
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Measurement Error
Types of errors: Systematic and random errors
Systematic error
Cause repeated readings to be in error by the same amount
Consistent, or fixed error component
May arise due to instrument short coming & environmental effects
Related to calibration errors and can be eliminated by correct calibration
Or human error such as consistent misreading and arithmetic error such as incorrect rounding off
Or by using an inadequate measurement methods
Example unjustified extrapolation of experimental data
Accuracy is related to such type of errors
Introduction to Instrumentation Engineering- Ch. 2 Analysis of Experimental Data

5. Measurement Error …. Random errors
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Measurement Error ….
Random errors
Due to unknown cause and occurs when all systematic errors have been accounted for
Caused by random electronic fluctuations in instruments, unpredictable behavior of the instrument, influences of friction, etc…
Random fluctuations usually follow certain statistical distribution
Treated by statistical methods
Characterized by positive and negative errors
Such errors are related to precision
Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

6. Measurement Error …. Systematic errors analysis can be divided into
Worst-cases analysis and RMS error analysis
Worst-case analysis: Let Qm be the measured quantity and Qt be true quantity
Error:
Relative error:
E.g., if the measured value is 10.1 when the true value is 10.0, the error is If the measured value is 9.9 when the true value is 10.0, the error is +0.1
Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

7. Error & Uncertainty Uncertainty
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Error & Uncertainty
Uncertainty
Since the true value cannot be known, the error of a measurement is also unknown
Thus, the closeness of the value obtained through a measurement to the true value is unknown
We are uncertain how well our measured value represents the true value
Uncertainty characterizes the dispersion of values
±Ea is the assigned uncertainty of Ea
Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

8. Error and Uncertainty …
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Error and Uncertainty …
Differentiate between error and uncertainty
Error indicate knowledge of the correct value
May be either positive or negative!
Uncertainty indicate lack of knowledge of the correct value or may be either positive or negative!
Is always a positive quantity, like standard deviation
Introduction to Instrumentation Engineering- Ch. 2 Analysis of Experimental Data

9. Combined Uncertainty – Commonsense Basis
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Combined Uncertainty – Commonsense Basis
Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

10. Combined Uncertainty …
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Combined Uncertainty …
Given a function
The RMS error is given as
And
Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

11. Uncertainty of Measurements
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Uncertainty of Measurements
Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

12. Overview Measurement error analysis Types of errors and uncertainty
Statistical analysis
Binomial and Gaussian distributions
Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

13. Statistical Analysis
Allows analytical determination of uncertainty of test result
Arithmetic mean (or most probable value) of n readings x1 to xn is given by
With a large sample, frequency distribution of the individual xi’s can be used to save time
If a particular value of xi occurs in the sample fj times, the mean value can be determined as
The sample frequency fj/n is an estimate of the probability Pj that x has the value of xj in the population sample
Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

14. Statistical Analysis …
Deviation xi-xm is the difference of all readings or observations from the mean reading
Is a good indicator of the uncertainty of the instrument
Average deviation: Sum of the absolute value of all deviations, i.e.,
Tends to zero and gives an indication of the precision of the instrument (low value shows that the instrument is highly precise)
Standard deviation: deviation from the mean & is given as
σ is called the population or biased standard deviation
Measure the extent of expected error in any observation
Variance: σ2
Introduction to Instrumentation Engineering- Ch. 2 Analysis of Experimental Data

15. Statistical Analysis …
Using the probability distribution Pj and noting that Pj=fj/n
For most distributions (both real and theoretical) met in statistical work, more than 94% of all observations in the population are within the interval xm  2
Generally, it is desirable to have about 20 observations in order to obtain reliable estimate of 
For smaller set of data, the expression for  modifies to
Called unbiased or sample standard deviation
Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

16. Cumulative Frequency Distribution
Sometimes the investigator is interested in estimating the proportion of the data whose values exceed some stated level or fall short of the level
E.g., for the random number, if the cumulative frequency distribution for drawing digits less than 3 is 0.75 and the cumulative frequency distribution for drawing digits >= 3 is 0.25
See the cumulative frequency distribution shown in the figure
Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

17. Overview Measurement error analysis Types of errors and uncertainty
Statistical analysis
Gaussian and Binomial distributions
Method of Least Squares
Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

18. Gaussian Distribution
Measurements will always have random errors
For a large number of data, these errors will have a normal distribution which follows
P(x) is the probability density function
It gives the probability that the data x will lie between x and x+dx
Is called the Gaussian or Normal error distribution
Xm is the mean and  is the standard deviation
Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

19. Gaussian Distribution …
Gaussian error distribution for σ=0.5 and 1 and xm=3
Probability density function has the property
Xm is the most probable reading
The value of the maximum probability density function is
Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

20. Gaussian Distribution …
The standard deviation is a measure of the width of the distribution curve about the mean
Smaller σ produces larger value of the maximum probability
For a measurement, this tends to go to more precision
The probability that a measurement will fall within a certain range x1 of the mean reading is given by
Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

21. Binomial Distribution
In many statistical analysis, the samples may consist of only two kinds of elements
E.g., odd or even, pass or fail, male or female, infested or free, dead or alive, etc.
We may be interested in the proportion, percentage, or number of data in one of the two classes
The head and tail case of a coin throw is very typical in this respect
The chances of having a head or a tail in one throw is 50% if the coin is not weighted
In other words the frequency of occurrence is the same for both heads and tails
Introduction to Instrumentation Engineering- Ch. 2 Analysis of Experimental Data

22. Binomial Distribution …
Supposing we toss the same coin twice (or toss two coins once), the outcomes could be any of the following:
H H H T T H T T
½ x ½ ½ x ½ ½ x ½ ½ x ½
¼ ¼ ¼ ¼
The chance of getting one H and one T is (¼ + ¼)= ½
Sum of probabilities is ¼+ ½+¼=1
For three throws, there will be 8 outcomes
HHH HHT HTT HTH TTH THT THH TTT
Each occurrence has a chance of (½ x ½ x ½ =⅛)
Introduction to Instrumentation Engineering- Ch. 2 Analysis of Experimental Data

23. Binomial Distribution …
3H only one → gives the chance of ⅛
2H and 1T-3 of them → gives ⅜
2T and 1H-3 of them → gives ⅜
3T only one → gives ⅛
The sum of the probabilities is ⅛+ ⅜ + ⅜ +⅛ = 1
To generalize, if the outcomes are successes (S) and failures (F) with probability of success being (p) and that of failure (q) where (p+q=1), for a sample size of N 2
Each of the success (events) can be determined from
where n is the number of success
Introduction to Instrumentation Engineering- Ch. 2 Analysis of Experimental Data

24. Binomial Distribution …
Requires the number of mutually exclusive ways in which the n successes and the (N-n) failures can be arranged
Statistically, this term is called the number of combinations of n letters out of N letters and is given by
The probability that n events will result in success stories is
The right one is a binomial expansion expression, hence the distribution called the binomial distribution
Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

25. Binomial Distribution …
E.g., for events of 0 and 1 designated by xj, mean of the binomial distribution is
Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

26. Overview Measurement error analysis Types of errors and uncertainty
Statistical analysis
Gaussian and Binomial distributions
Method of Least Squares
Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

27. Method of Least Squares
In the operation of an instrument, input parameter is varied over some range
Could be in increments or decrements
Happens during calibration or measurement
Least square can be applied to determine an equation for a measured data
Used to fit the data into a line (cure) to give a working relation between input and output
This relation will help to determine the characteristics of the instrument
Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

28. Method of Least Squares …
Example: Linear Least Square Analysis (LLSA)
Suppose xi and yi be the input and measured values, respectively such that the data points (x1 , y1), (x2 , y2), ….. (xn , yn) are obtained
If the expected straight line is of the form
y = mx + b
where m is the slope and b is the intercept
The error, which is the difference between the actual and measured data, summed for all points is given as
Minimizing S using
Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

29. Method of Least Squares …
Will give
And
Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

30. Method of Least Squares …
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Method of Least Squares …
Example: Assume that the input and output are related by a second order equation
y = b2x2 + b1x +b0
The error will take the form
Minimizing the error with respect to b0, b1, and b2 yields
n = total number of data points
Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

31. Method of Least Squares …
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Method of Least Squares …
Assignment: The iron losses (L) in a ferromagnetic material, which is used to construct a transformer, vary with frequency (f) of the supply driving the transformer
For a particular transformer, these losses were determined at various frequencies with a constant flux density in the ferromagnetic material
Assume that the iron losses have a general form
Using LLSA, determine the constants A and B.
Frequency (Hz)
1100
1400
1700
2000
Iron losses (mW) 46 62 94
122
Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

32. Method of Least Squares …
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Method of Least Squares …
Example: Consider data of high school versus college GPA, given as x and y, respectively. Compute the equation of linear least square regression line.
Student x y x2 y2 xy 1
2.0
1.6
4.00
2.56
3.20 2
2.2
4.84
4.40 3
2.6
1.8
6.76
3.24
4.68 4
2.7
2.8
7.29
7.84
7.56 5
2.1
4.41
5.88 6
3.1
9.61
6.20 7
2.9
8.41
7.54 8
3.2
10.24
7.04 9
3.3
10.89
8.58 10
3.6
3.0
12.96
9.00
10.80
Totals
28.4
22.7
82.84
53.41
65.88
Introduction to Instrumentation Engineering- Ch. 2 Analysis of Experimental Data