44
LECTURER PROF.Dr. DEMIR BAYKA AUTOMOTIVE ENGINEERING LABORATORY I
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LECTURER
PROF.Dr. DEMIR BAYKA
AUTOMOTIVE ENGINEERING LABORATORY
I
STATISTICAL TREATMENT OF EXPERIMENTAL DATA
DISCRETE FREQUENCY DISTRIBUTIONS
x3 x1
x6
x7
x8
x10
x2
x9
x5 x4
13 14 15 16 17 18 19
4
3
2
1
0
Frequencynj 0.4
0.3
0.2
0.1
0.0
RelativeFrequency
fj
x3 x1
x6
x7
x8
x10
x2
x9
x5 x4
13 14 15 16 17 18 19
4
3
2
1
0
Frequencynj 0.4
0.3
0.2
0.1
0.0
RelativeFrequency
fj
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
14 16 13 19 18 14 14 15 18 15
FREQUENCY F( nj )
IS THE NUMBER OF OCCURRENCE OF THE jth MEASUREMENT VALUE
j 1 2 3 4 5 6 7value 13 14 15 16 17 18 19nj 1 3 2 1 0 2 1
j 1 2 3 4 5 6 7value 13 14 15 16 17 18 19nj 1 3 2 1 0 2 1
x3 x1
x6
x7
x8
x10
x2
x9
x5 x4
13 14 15 16 17 18 19
4
3
2
1
0
Frequencynj 0.4
0.3
0.2
0.1
0.0
RelativeFrequency
fj
RELATIVE FREQUENCY fj
IS THE RELATIVE VALUES OF NUMBER OF OCCURRENCES WITH RESPECT TO TOTAL NUMBER OF OCCURRENCES
n
nf j
j 1fm
1jj
m
1jjnn&
RELATIVE FREQUENCY fj
THERE ARE 7 GROUPS ie m = 7
j 1 2 3 4 5 6 7
value 13 14 15 16 17 18 19
fj 0.1 0.3 0.2 0.1 0.0 0.2 0.1
FREQUENCY GRAPH
x3 x1
x6
x7
x8
x10
x2
x9
x5 x4
13 14 15 16 17 18 19
4
3
2
1
0
Frequencynj 0.4
0.3
0.2
0.1
0.0
RelativeFrequency
fj
MEASURES OF CENTRAL TENDENCY
ARITHMETIC MEAN
n
1iix
n
1x
IT PROVIDES THE BEST ESTIMATE OF AN UNBIASED DISTRIBUTION OF DATA
MEASURES OF CENTRAL TENDENCY
MEDIAN
IT IS THE VALUE AT THE MIDDLE POSITION OF A DISTRIBUTION OF DATA
IT IS USUALLY USED WHEN THE DISTRIBUTION IS BIASED
MEASURES OF CENTRAL TENDENCY
MODE
IT IS THE VALUE HAVING THE HIGHEST FREQUENCY
IN THE SAMPLE DISTRIBUTION
GEOMETRIC MEAN (Log - Mean)
n/1n
1iig xx
n
1iig )xlog(
n
1)xlog(
IT IS IMPORTANT WHEN DEALING WITH RATIOS OR PERCENTAGES
HARMONIC MEAN
n
1iih )x/1(nx
QUADRATIC MEAN
(ROOT - MEAN - SQUARE )
n
1i
2irms x
n
1x
REPEATED MEASUREMENTS
TIMEt = 0.5 s
8.60
8.25
8.30
8.35
8.40
8.45
8.50
8.55
8.65
8.70
8.20
THIS IS ASSUMEDTO REPRESENT THE TRUE VALUE AS BEST AS POSSIBLE
8.45TAKE
AVERAGE
4 8.495 8.416 8.587 8.438 8.53
3 8.48
9 8.6510 8.40
20 8.35
1 8.682 8.25
19 8.5618 8.2817 8.23
13 8.31
11 8.4812 8.37
16 8.5015 8.5014 8.52
START SAMPLING END SAMPLING
RATE OFSAMPLING5 ms (200 kHz)
MEASURES OF DISPERSION OF DATA
VARIANCE(MEAN SQUARE DEVIATION )
n
1i
2i
2 )xx(n
1VAR
0.01497
8.45-8.358.45-8.568.45-8.288.45-8.23
8.45-8.58.45-8.58.45-8.528.45-8.31
8.45-8.378.45-8.488.45-8.48.45-8.65
8.45-8.538.45-8.438.45-8.588.45-8.41
8.45-8.498.45-8.488.45-8.258.45-8.68
20
1
2222
2222
2222
2222
2222
2
MEASURES OF DISPERSION OF DATA
STANDARD DEVIATION
0.1223520.01497
)x()x()xx(n
1 22i
n
1i
2i
REPEATED MEASUREMENTS
TIMEt = 0.5 s
8.60
8.25
8.30
8.35
8.40
8.45
8.50
8.55
8.65
8.70
8.20
8.45TAKE
AVERAGE
8.45 + = 8.57
8.45 - = 8.33
65%
= 0.122352
MEASURES OF DISPERSION OF DATA
RANGE
IT IS THE DIFFERENCE BETWEEN THE LARGEST AND SMALLEST VALUES OF THE ENTIRE SET OF DATA
MEASURES OF DISPERSION OF DATA
AVERAGE DEVIATION
n
1i
xi
xn
1.A.D
UNBIASED ESTIMATES
Population or Universe
Mean: S.D.:
Random Sample (x1, x2, … , xn)
UNBIASED ESTIMATES
Population or Universe
Mean:
S.D.:
Random Sample (x1, x2, … , xn)
A) THE SAMPLE MEAN x IS THE BEST AVAILABLE ESTIMATE OF THE UNKNOWN MEAN OF THE UNIVERSE
UNBIASED ESTIMATES
Population or Universe
Mean:
S.D.:
Random Sample (x1, x2, … , xn)
A) THE BEST AVAILABLE ESTIMATE OF THE UNKNOWN STANDARD DEVIATION OF THE UNIVERSE IS GIVEN BY
22i
n
1i
2i )x()x(
1n
n)xx(
1n
1
22i
n
1i
2i )x()x(
1n
n)xx(
1n
1
THE USE OF THIS EXPRESSION BECOMES IMPORTANT ESPECIALLY WHEN n IS SMALL
FOR LARGE VALUES OF n sample
HOWEVER, > sample ALWAYS
C) IF MORE THAN ONE ( SAY m ) EQUAL-SIZED RANDOM SAMPLES ARE DRAWN FROM THE SAME UNIVERSE, THEN THEIR RESPECTIVE MEANS AND STANDARD DEVIATIONS ARE EXPECTED TO BE EQUAL TO EACH OTHER
m21
m21
.....
x.....xx
Population or Universe
Sample 2
Sample 1
Sample m
m21
m21
.....
x.....xx
STANDARD ERROR OF THE MEAN
nx
THIS QUANTITY REPRESENTS THE STANDARD DEVIATION OF
x FROM
REPEATED MEASUREMENTS
TIMEt = 0.5 s
8.60
8.25
8.30
8.35
8.40
8.45
8.50
8.55
8.65
8.70
8.20
8.45
8.45 + = 8.57
8.45 - = 8.33
65%
= 0.122352
REPEATED MEASUREMENTS
TIMEt = 0.5 s
8.60
8.25
8.30
8.35
8.40
8.45
8.50
8.55
8.65
8.70
8.20
8.45
= 0.122352
0.02735920
122352.0x
03.0
8.48
8.42THE TRUE VALUEIS IN THIS RANGEWITH 68%CONFIDENCE
STANDARD ERROR OF THE STANDARD DEVIATION
2n2x
THIS QUANTITY REPRESENTS THE STANDARD DEVIATION OF x FROM
REPEATED MEASUREMENTS
TIMEt = 0.5 s
8.60
8.25
8.30
8.35
8.40
8.45
8.50
8.55
8.65
8.70
8.20
8.45
= 0.122352
0.02735920
122352.0x
03.0
0.019346
2
027359.0
2n2x
0.0080130.0193460.027359
0.0467040.0193460.027359
Lx
Hx
0.030.027359x
CONTINUOUS DISTRIBUTIONS
IN ACTUAL EXPERIMENTS VALUES WILL BE LESS DISCRETE
23.26 , 25.12 , etc
CONTINUOUS DISTRIBUTIONS
IF WE HAD A SET OF 100 DATA VALUES SUCH AS 23.26 , 25.12 ... , etc THEN THE FREQUENCY GRAPH WOULD PROBABLY HAVE VERY FEW VALUES THAT WERE THE SAME
CONTINUOUS DISTRIBUTIONS
CONTINUOUS DISTRIBUTIONS
THE ONLY APPARENT MEANINGFUL QUANTITY APPEARS TO BE THE DENSITY OF THE “DOTS”
CONTINUOUS DISTRIBUTIONS
16
LET US DIVIDE THE DATA BY 5.0
INCREMENTS
CONTINUOUS DISTRIBUTIONSNOW LET US COUNT HOW MANY DATA POINTS ARE BETWEEN 22.51 AND 23.50
16
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
14 16 13 19 18 14 14 15 18 15
x3 x1
x6
x7
x8
x10
x2
x9
x5 x4
13 14 15 16 17 18 19
4
3
2
1
0
Frequencynj 0.4
0.3
0.2
0.1
0.0
RelativeFrequency
fj
IF MORE MEASUREMENTS WITH A MORE ACCURATE DEVICE WERE TAKEN
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
14.21 16.36 13.16 18.74 17.59 14.43 14.02 14.77 18.01 15.16
13 14 15 16 17 18 19
0.2
0.1
0.0
Relative Frequency, fj
AND IF THE DATA WERE INCREASED
13 14 15 16 17 18 19
0.10
0.05
0.00
Relative Frequency, fj
13 14 15 16 17 18 19
0.10
0.05
0.00
Relative Frequency, fj
13 14 15 16 17 18 19
0.04
0.02
0.00
Relative Frequency, fj
0.06
0.08 Envelope
THE INTERVAL MUST BE CHOSEN
* LARGE ENOUGH TO BEMEANINGFUL
* SMALL ENOUGH TO GIVE DETAIL
N = 5 log n for large nN = 1 + 3.3 log n for n<25 Sturges rulewhere n is the number of data points and N issuggested number of class intervals.
