1. LECTURE 3: ANALYSIS OF EXPERIMENTAL DATA
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 Measures of dispersion Introduction 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 = (62.18)

10. Contents Parameter estimation Introduction Measures of dispersion
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. Level of Significance (%)
Table z
Confidence Interval
Confidence Level (%)
Level of Significance (%)
3.30
99.9
0.1
3.0
99.7
0.3
2.57
99.0
1.0
2.0
95.4
4.6
1.96
95.0
5.0
1.65
90.0
10.0
68.3
31.7
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 Criterion for rejection questionable data points Introduction
Measures of dispersion
Parameter estimation
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, , and
= 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 = should be discarded.
Recalculate S and to obtain 0.05 and respectively.
Hence for n = 8, and
so remaining data stay.
Thompson’s t table

23. Contents Correlation of experimental data Introduction
Measures of dispersion
Parameter estimation
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 Determine the linear correlation between T and V
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:
T(0C) 20 30 40 50 60 75
100
V(mV)
1.02
1.53
2.05
2.55
3.07
3.56
4.05
Determine the linear correlation between T and V

35. Tabulate the data using this table:
Solution:
Tabulate the data using this table:
rxy=

36. The following measurements were obtained in the calibration of
Another example
The following measurements were obtained in the calibration of
a pressure transducer:
Voltage
DP H2O
0.31
1.96
0.65
4.20
0.75
4.90
0.85
5.48
0.91
5.91
1.12
7.30
1.19
7.73
1.38
9.00
1.52
9.90
Determine the best fit
straight line
Find the coefficient of
determination for the
best fit

37. Y=6.56x-0.06

38. r2= 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