LECTURE 3: ANALYSIS OF EXPERIMENTAL DATA

1. MEASUREMENT AND INSTRUMENTATION BMCC 3743 LECTURE 3: ANALYSIS OF EXPERIMENTAL DATA Mochamad Safarudin Faculty of Mechanical Engineering, UTeM 2010

2. Contents • Introduction • Measures of dispersion • Parameter estimation • Criterion for rejection questionable data points • Correlation of experimental data

3. Introduction • Needed in all measurements with random inputs, e.g. random broadband sound/noise • Tyre/road noise, rain drops, waterfall • Some important terms are: • Random variable (continuous or discrete), histogram, bins, population, sample, distribution function, parameter, event, statistic, probability.

4. Terminology • Population : the entire collection of objects, measurements, observations and so on whose properties are under consideration • Sample: a representative subset of a population on which an experiment is performed and numerical data are obtained

5. Contents • Introduction • Measures of dispersion • Parameter estimation • Criterion for rejection questionable data points • Correlation of experimental data

6. Measures of dispersion =>Measures of data spreading or variability • Deviation (error) is defined as • Mean deviation is defined as • Population standard deviation is defined as

7. Measures of dispersion • Sample standard deviation is defined as • is used when data of a sample are used to estimate population std dev. • Variance is defined as

8. Exercise • Find the mean, median, standard deviation and variance of this measurement: 1089, 1092, 1094, 1095, 1098, 1100, 1104, 1105, 1107, 1108, 1110, 1112, 1115

9. Answer to exercise • Mean = 1103 (1102.2) • Median = 1104 • Std deviation = 5.79 (7.89) • Variance = 33.49 (62.18)

10. Contents • Introduction • Measures of dispersion • Parameter estimation • Criterion for rejection questionable data points • Correlation of experimental data

11. Parameter estimation Generally, • Estimation of population mean, is sample mean, . • Estimation of population standard deviation, is sample standard deviation, S.

12. Interval estimation of the population mean • Confidence interval is the interval between to , where is an uncertainty. • Confidence level is the probability for the population mean to fall within specified interval:

13. Interval estimation of the population mean • Normally referred in terms of , also called level of significance, where confidence level • If n is sufficiently large (> 30), we can apply the central limit theorem to find the estimation of the population mean.

14. Central limit theorem • If original population is normal, then distribution for the sample means’ is normal (Gaussian) • If original population is not normal and n is large, then distribution for sample means’ is normal • If original population is not normal and n is small, then sample means’ follow a normal distribution only approximately.

15. Normal (Gaussian) distribution • When n is large, where • Rearranged to get • Or with confidence level

16. Area under 0 to z

17. Student’s t distribution • When n is small, where • Rearranged to get • Or with confidence level t table

18. Interval estimation of the population variance • Similarly as before, but now using chi-squared distribution, , (always positive) where

19. Interval estimation of the population variance • Hence, the confidence interval on the population variance is Chi squared table

20. Contents • Introduction • Measures of dispersion • Parameterestimation • Criterion for rejection questionable data points • Correlation of experimental data

21. Criterion for rejection questionable data points • To eliminate data which has low probability of occurrence => use Thompson test. • Example: Data consists of nine values, Dn = 12.02, 12.05, 11.96, 11.99, 12.10, 12.03, 12.00, 11.95 and 12.16. • = 12.03, S = 0.07 • So, calculate deviation:

22. Criterion for rejection questionable data points • From Thompson’s table, when n = 9, then • Comparing with where then D9 = 12.16 should be discarded. • Recalculate S and to obtain 0.05 and 12.01 respectively. • Hence forn = 8, and so remaining data stay. Thompson’s t table

23. Contents • Introduction • Measures of dispersion • Parameterestimation • Criterion for rejection questionable data points • Correlation of experimental data

24. Correlation of experimental data • Correlation coefficient • Least-square linear fit • Linear regression using data transformation

25. A) Correlation coefficient • Case I: Strong, linear relationship between x and y • Case II: Weak/no relationship • Case III: Pure chance => Use correlation coefficient, rxy to determine Case III

26. Linear correlation coefficient • Given as where • +1 means positive slope (perfectly linear relationship) • -1 means negative slope (perfectly linear relationship) • 0 means no linear correlation

27. Linear correlation coefficient • In practice, we use special Table (using critical values of rt) to determine Case III. • If from experimental value of |rxy|is equal or more than rt as given in the Table, then linear relationship exists. • If from experimental value of |rxy|is less than rt as given in the Table, then only pure chance => no linear relationship exists.

28. B) Least-square linear fit To get best straight line on the plot: • Simple approach: ruler & eyes • More systematic approach: least squares • Variation in the data is assumed to be normally distributed and due to random causes • To get Y = ax + b, it is assumed that Y values are randomly vary and x values have no error.

29. Least-square best fit • For each value of xi, error for Y values are • Then, the sum of squared errors is

30. Least-square best fit • Minimising this equation and solving it for a & b, we get

31. Least-square best fit • Substitute a & b values into Y = ax + b, which is then called the least-squares best fit. • To measure how well the best-fit line represents the data, we calculate the standard error of estimate, given by where Sy,x is the standard deviation of the differences between data points and the best-fit line. Its unit is the same as y.

32. Coefficient of determination • …Is another good measure to determine how well the best-fit line represents the data, using • For a good fit, must be close to unity.

33. C) Linear regression using data transformation • For some special cases, such as • Applying natural logarithm at both sides, gives where ln(a) is a constant, so ln(y) is linearly related to x.

34. Example • Thermocouples are usually approximately linear devices in a limited range of temperature. A manufacturer of a brand of thermocouple has obtained the following data for a pair of thermocouple wires: Determine the linear correlation between T and V

35. Solution: Tabulate the data using this table:

36. Another example The following measurements were obtained in the calibration of a pressure transducer: • Determine the best fit • straight line • Find the coefficient of • determination for the • best fit

37. Y=6.56x-0.06

38. From the result before we can find coeff of determination r2 by tabulating the following values r2=

39. Next Lecture Experimental Uncertainty Analysis End of Lecture 3