LECTURE 3: ANALYSIS OF EXPERIMENTAL DATA

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MEASUREMENT AND INSTRUMENTATION BMCC 3743. LECTURE 3: ANALYSIS OF EXPERIMENTAL DATA. Mochamad Safarudin Faculty of Mechanical Engineering, UTeM 2010. Contents. Introduction Measures of dispersion Parameter estimation Criterion for rejection questionable data points

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MEASUREMENT AND INSTRUMENTATION

BMCC 3743

LECTURE 3: ANALYSIS OF EXPERIMENTAL DATA

Faculty of Mechanical Engineering, UTeM

2010

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

• Mean = 1103 (1102.2)
• Median = 1104
• Std deviation = 5.79 (7.89)
• Variance = 33.49 (62.18)
Contents
• Introduction
• Measures of dispersion
• Parameter estimation
• Criterion for rejection questionable data points
• Correlation of experimental data
Parameter estimation

Generally,

• Estimation of population mean, is sample mean, .
• Estimation of population standard deviation, is sample standard deviation, S.
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:
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.
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.
Normal (Gaussian) distribution
• When n is large,

where

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

where

• Rearranged to get
• Or with confidence level

t table

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

where

Interval estimation of the population variance
• Hence, the confidence interval on the population variance is

Chi squared table

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

Contents
• Introduction
• Measures of dispersion
• Parameterestimation
• Criterion for rejection questionable data points
• Correlation of experimental data
Correlation of experimental data
• Correlation coefficient
• Least-square linear fit
• Linear regression using data transformation
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

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
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.
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.
Least-square best fit
• For each value of xi, error for Y values are
• Then, the sum of squared errors is
Least-square best fit
• Minimising this equation and solving it for a & b, we get
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.

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.
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.

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

Solution:

Tabulate the data using this table:

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
Next Lecture

Experimental Uncertainty Analysis

End of Lecture 3