Introduction to Instrumentation Engineering

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

2. 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 • 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 • 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 -0.1. 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 • 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 … • 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 Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

10. 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 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 toxnis 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 probabilityPjthat 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 … • 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 … • 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 … • 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. Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

32. 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. Introduction to Instrumentation Engineering- Ch. 2 Analysis of Experimental Data