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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
- 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

Answer to exercise

- 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.

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

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