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 …. 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

6. 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

7. 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

8. 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

9. Overview Measurement error analysis Types of errors and uncertainty
Random Error Analysis
Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

10. 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

11. 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

12. 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

13. 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

14. 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

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

16. 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

17. 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

18. 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

19. Contnd.
Almost all random observation have a probability density distribution P(x) which is a well known Analytical form
Where d is the deviation and
h is a parameter representing a measure of precision
The relationship between h, σ and D for a Gaussian Distribution
The probable error has been used in experimental work
Probable Error=0.675 σ
Instrum. & Control Eng. for Energy Systems - Ch. 2 Analysis of Experimental Data

20. Class Exercise
The measurement of Resistance of a 100 ohm resistor give the following results 101.2,101.7,101.3,101.0,101.5,101.3,101.2,101.4,101.3 and ohms Assuming that the random errors are present Determine a) The arithmetic mean of the data b) The Deviation of the data and the sum of deviation of the data. c)The Average deviation of the data d) The Standard deviation and e) The Probable Error
Instrum. & Control Eng. for Energy Systems - Ch. 2 Analysis of Experimental Data

21. Class Exercise
A set of Independent current measurement was taken by six observers and recorded as 12.8mA,12.2mA,12.5mA,12.9mA,12.6mA and 12.4mA Determine a) The arithmetic mean of the data b) The Deviation of the data and the sum of deviation of the data. c)The Average deviation of the data d) The Standard deviation and e) The Probable Error
Instrum. & Control Eng. for Energy Systems - Ch. 2 Analysis of Experimental Data

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

23. Random Error Analysis of Scattered Data
If the scatter of Data point is not too great, it is relatively simple to draw the input and output using straight line through the data points.
If the Data points are widely scattered the Linear Least square algorithm or Regression Analysis can be used
The LLSA is a method for fitting the best curve of a pre-determined form through a series of experimental data points Or
The LLSA is also known as the “minimum square error method for the best curve fitting”
Instrum. & Control Eng. for Energy Systems - Ch. 2 Analysis of Experimental Data

24. 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

25. 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

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

27. 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