LECTURE 3: ANALYSIS OF EXPERIMENTAL DATA

Mochamad Safarudin

Faculty of Mechanical Engineering, UTeM

2010

MEASUREMENT AND INSTRUMENTATIONBMCC 3743

Introduction Measures of dispersion Parameter estimation Criterion for rejection questionable data

points Correlation of experimental data

2

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.

3

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

4

Introduction

Measures of dispersion Parameter estimation Criterion for rejection questionable data

points Correlation of experimental data

5

Deviation (error) is defined as

Mean deviation is defined as

Population standard deviation is defined as

xxd ii

n

i

i

n

dd

1

6

=>Measures of data spreading or variability

N

i

i

N

x

1

2

Sample standard deviation is defined as

◦ is used when data of a sample are used to estimate population std dev.

Variance is defined as

n

i

i

n

xxS

1

2

1

sampleaforS

or

populationthefor

2

2

7

Find the mean, median, standard deviation and variance of this measurement:1089, 1092, 1094, 1095, 1098, 1100, 1104, 1105,

1107, 1108, 1110, 1112, 1115

8

Mean = 1103 (1102.2) Median = 1104 Std deviation = 5.79 (7.89) Variance = 33.49 (62.18)

9

Introduction Measures of dispersion

Parameter estimation Criterion for rejection questionable data

points Correlation of experimental data

10

Generally,Estimation of population mean, is sample mean, .

Estimation of population standard deviation, is sample standard deviation, S.

x

11

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:

12

xxP

x x

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.

13

1

1. If original population is normal, then distribution for the sample means’ is normal (Gaussian)

2. If original population is not normal and n is large, then distribution for sample means’ is normal

3. If original population is not normal and n is small, then sample means’ follow a normal distribution only approximately.

14

When n is large,

where

Rearranged to get

Or with confidence level

15

1/

2/2/ zn

xzP

n

xz

/

nzx

2/ 1

12/2/n

zxn

zxP

16

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

1.0 68.3 31.7

Area under 0 to z

When n is small,

where

Rearranged to get

Or with confidence level

17

1/

2/2/ tnS

xtP

nS

xt

/

12/2/n

Stx

n

StxP

n

Stx 2/ 1

t table

Similarly as before, but now using chi-squared distribution, , (always positive)

where

18

2

11 2

2/,2

22

2/1, vv

SnP

2

22 1

S

n

Hence, the confidence interval on the population variance is

19

2

2/1,

22

22/,

2 11

vv

SnSn

Chi squared table

Introduction Measures of dispersion Parameter estimation

Criterion for rejection questionable data points

Correlation of experimental data

20

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:

21

08.003.1295.11

13.003.1216.12

2

arg1

DD

DD

smallest

estl

D

From Thompson’s table, when n = 9, then

Comparing withwhere then D9 = 12.16 should be discarded.

Recalculate S and to obtain 0.05 and 12.01 respectively.

Hence for n = 8, andso remaining data stay.

22

777.1

12.077.107.0 S ,13.01 ,1 S

D

749.1 ,09.0S

Thompson’s ble

Introduction Measures of dispersion Parameter estimation Criterion for rejection questionable data

points

Correlation of experimental data

23

A) Correlation coefficientB) Least-square linear fitC) Linear regression using data

transformation

24

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

25

Given as

where +1 means positive slope (perfectly linear

relationship) -1 means negative slope (perfectly linear

relationship) 0 means no linear correlation

26

2/1

1 1

22

1

n

i

n

iii

i

n

ii

xy

yyxx

yyxxr

11 xyr

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.

27

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.

28

For each value of xi, error for Y values are

Then, the sum of squared errors is

29

iii yYe

n

iii

n

iii ybaxyYE

1

2

1

2

Minimising this equation and solving it for a & b, we get

30

22

2

22

ii

iiiii

ii

iiii

xxn

yxxyxb

xxn

yxyxna

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.

31

2111

2

,

n

yxaybyS i

xy

…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.

32

2

2

2 1yy

ybaxr

i

ii

2r

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.

33

bxaey

abxy lnln

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: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

Solution:

Tabulate the data using this table:

2/1

1 1

22

1

n

i

n

iii

i

n

ii

xy

yyxx

yyxxr rxy= 0.980392

No x (0C) y(mV)1 20 1.02 -33.57 1127.04 -1.53 2.33 51.272 30 1.53 -23.57 555.61 -1.02 1.03 23.983 40 2.05 -13.57 184.18 -0.50 0.25 6.754 50 2.55 -3.57 12.76 0.00 0.00 -0.015 60 3.07 6.43 41.33 0.52 0.27 3.366 75 3.56 21.43 459.18 1.01 1.03 21.707 100 4.05 46.43 2155.61 1.50 2.26 69.78

53.572.55

4535.71 7.17 176.82

xy

xx i yy i )( xx i )( yy i 2)( xx i 2yy i

Another example

The following measurements were obtained in the calibration ofa pressure transducer:

Voltage P 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

a. Determine the best fit straight line

b. Find the coefficient ofdetermination for the best fit

xi xi2 yi xiyi yi

2

0.31 0.0961 1.96 0.6076 3.84160.65 0.4225 4.2 2.73 17.640.75 0.5625 4.9 3.675 24.010.85 0.7225 5.48 4.658 30.03040.91 0.8281 5.91 5.3781 34.92811.12 1.2544 7.3 8.176 53.291.19 1.4161 7.73 9.1987 59.75291.38 1.9044 9 12.42 811.52 2.3104 9.9 15.048 98.01

sum () 8.68 9.517 56.38 61.8914 402.503

22

2

22

ii

iiiii

ii

iiii

xxn

yxxyxb

xxn

yxyxna

a= 6.560646

b= -0.062934

Y=6.56x-0.06

2

2

2 1yy

ybaxr

i

ii

0.999926r2=

xi yi (Yi-yi)2 (yi-y)2

0.31 1.96 0.000118 18.530.65 4.2 0.000002 4.260.75 4.9 0.001802 1.860.85 5.48 0.001130 0.620.91 5.91 0.000008 0.131.12 7.3 0.000225 1.071.19 7.73 0.000203 2.151.38 9 0.000085 7.481.52 9.9 0.000086 13.22

sum () 0.003659 49.31

From the result before we can find coeff of determination r2

by tabulating the following values

Experimental Uncertainty Analysis

End of Lecture 3

39