LECTURER PROF.Dr. DEMIR BAYKA AUTOMOTIVE ENGINEERING LABORATORY I.

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


MEDIAN

IT IS THE VALUE AT THE MIDDLE POSITION OF A DISTRIBUTION OF DATA

IT IS USUALLY USED WHEN THE DISTRIBUTION IS BIASED


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


STANDARD DEVIATION

0.1223520.01497

)x()x()xx(n

1 22i

n

1i

2i


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


RANGE

IT IS THE DIFFERENCE BETWEEN THE LARGEST AND SMALLEST VALUES OF THE ENTIRE SET 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


Mean:

S.D.:


A) THE SAMPLE MEAN x IS THE BEST AVAILABLE ESTIMATE OF THE UNKNOWN MEAN OF THE UNIVERSE

UNBIASED ESTIMATES


Mean:

S.D.:


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


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


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


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


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


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



THE ONLY APPARENT MEANINGFUL QUANTITY APPEARS TO BE THE DENSITY OF THE “DOTS”


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


13 14 15 16 17 18 19

0.10

0.05

0.00


13 14 15 16 17 18 19

0.04

0.02

0.00


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.

LECTURER PROF.Dr. DEMIR BAYKA AUTOMOTIVE ENGINEERING LABORATORY I.

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