MEASUREMENT AND INSTRUMENTATION
BMCC 3743
LECTURE 3: ANALYSIS OF
EXPERIMENTAL DATA
Mochamad Safarudin
Faculty of Mechanical Engineering, UTeM
2010
 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
=>Measures of data spreading or variability

Deviation (error) is defined as
di  xi  x

Mean deviation is defined as
n
d 
di
n
Population standard deviation is defined
as
i 1


N
xi   2
i 1
N

6

Sample standard deviation is defined as
2
n

xi  x 
S 
i 1 n  1
◦ is used when data of a sample are used to
estimate population std dev.

Variance is defined as
 2 for the population
or
S 2 for a sam ple
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, x .
 Estimation of population
standard deviation, is sample
standard deviation, S.
11


Confidence interval is the interval between
, where  is an uncertainty.
x   to x  
Confidence level is the probability for the
population mean to fall within specified
interval:
Px      x   
12


Normally referred in terms of  , also called
level of significance, where
confidence level  1  
If n is sufficiently large (> 30), we can apply
the central limit theorem to find the
estimation of the population mean.
13
1.
2.
3.
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.
14

When n is large, P z / 2  x    z / 2   1  


/ n


where z  x  
/ n

Rearranged to get

 

P  x  z / 2
   x  z / 2
  1
n
n

 Or
 with confidence level
  x  z / 2
1
n
15
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
16

When n is small, P  t / 2  x    t / 2   1  


S/ n


where t  x  
S/ n

Rearranged to get
S
S 

P  x  t / 2
   x  t / 2
  1
n
n

S with confidence level 1  
 Or   x  t
 /2
n
t table
17

Similarly as before, but now using chi2
squared distribution, , (always positive)
where
 2

S2
2
P   v ,1 / 2  n  1 2   v , / 2   1  



 2  n  1
S2
2
18

Hence, the confidence interval on the
population variance is
n 1S 2   2  n 1S 2
v2, / 2
v2,1 / 2
Chi squared table
19

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.
D = 12.03, S = 0.07
So, calculate deviation:
1  Dl argest  D  12.16  12.03  0.13
 2  Dsmallest  D  11.95  12.03  0.08
21




From Thompson’s  table, when n = 9, then
  1.777
Comparing S  0.07 1.77  0.12 with 1  0.13,
where 1  S , then D9 = 12.16 should be
discarded.
Recalculate S and D to obtain 0.05 and 12.01
respectively.
Hence for n = 8,   1.749
so remaining data stay.
and S  0.09,
Thompson’s  table
22

Introduction
Measures of dispersion

Parameter estimation


Criterion for rejection questionable data
points
 Correlation
data
of experimental
23
A)
B)
C)
Correlation coefficient
Least-square linear fit
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
n
rxy 
 x  x  y  y 
i 1
i
i

2
2




x

x
y

y

i
 i

i 1
 i 1

n
n
1/ 2
where 1  rxy  1
+1 means positive slope (perfectly linear
relationship)
-1 means negative slope (perfectly linear
relationship)
0 means no linear correlation
26



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
ei  Yi  yi
n
n
2




E   Yi  yi  axi  b  yi
i 1
2
i 1
29

Minimising this equation and solving it for a
& b, we get
a
b
n xi yi   xi  yi 
n xi2   xi 
2
2
x
 i  yi   xi  xi yi 
n x   xi 
2
i
2
30


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
2
S y,x 
y
i
 b y1  a x1 y1
n2
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

…Is another good measure to determine how
well the best-fit line represents the data,
using
2
r
2
ax  b  y 

 1
 y  y
i
i
2
i

For a good fit,r 2 must be close to unity.
32

For some special cases, such as
y  aebx

Applying natural logarithm at both sides,
gives
ln y   bx  lna 
where ln(a) is a constant, so ln(y) is linearly
related to x.
33

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:
x (0C)
No
1
2
3
4
5
6
7
x
y
x
y(mV)
20
30
40
50
60
75
100
53.57
1.02
1.53
2.05
2.55
3.07
3.56
4.05
i

x
( x
i
-33.57
-23.57
-13.57
-3.57
6.43
21.43
46.43
1127.04
555.61
184.18
12.76
41.33
459.18
2155.61
 y  y i  y 2
-1.53
2.33
-1.02
1.03
-0.50
0.25
0.00
0.00
0.52
0.27
1.01
1.03
1.50
2.26
4535.71
7.17
 x )
2
y
i
( x
i

x )
( y

y )
51.27
23.98
6.75
-0.01
3.36
21.70
69.78
2.55

176.82
n
rxy 
i
 x  x  y  y 
i 1
i
i
n
n
2
2




x

x
y

y

i
 i

i 1
 i 1

1/ 2
rxy=
0.980392
Another example
The following measurements were obtained in the calibration of
a pressure transducer:
a. Determine the best fit
straight line
b. Find the coefficient of
determination for the
best fit
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
xi2
0.0961
0.4225
0.5625
0.7225
0.8281
1.2544
1.4161
1.9044
2.3104
9.517
xi
0.31
0.65
0.75
0.85
0.91
1.12
1.19
1.38
1.52
8.68
sum ()
a
n xi yi   xi  yi 
n xi2   xi 
2
 x  y   x  x y 
b
n x   x 
2
i
i
i
2
i
i
2
i
i
yi
1.96
4.2
4.9
5.48
5.91
7.3
7.73
9
9.9
56.38
xiyi
0.6076
2.73
3.675
4.658
5.3781
8.176
9.1987
12.42
15.048
61.8914
yi2
3.8416
17.64
24.01
30.0304
34.9281
53.29
59.7529
81
98.01
402.503
a=
6.560646
b=
-0.062934
Y=6.56x-0.06
From the result before we can find coeff of determination r2
by tabulating the following values
xi
yi
0.31
0.65
0.75
0.85
0.91
1.12
1.19
1.38
1.52
sum ()
1.96
4.2
4.9
5.48
5.91
7.3
7.73
9
9.9
(Yi-yi)2
0.000118
0.000002
0.001802
0.001130
0.000008
0.000225
0.000203
0.000085
0.000086
0.003659
(yi-y)2
18.53
4.26
1.86
0.62
0.13
1.07
2.15
7.48
13.22
49.31
ax  b  y 

 1
 y  y
2
r
2
i
i
2
i
r2=
0.999926
Experimental Uncertainty Analysis
End of Lecture 3
39