Statistics and DOE Latest

8/4/2019 Statistics and DOE Latest

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Statistics and DOE

Mayank


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MeaneanMedianedianModeode

( )asures of dispersion spread of dataVarianceariance

tandard deviationtandard deviationoefficient of variationoefficient of variation

( )s of central tendency central position of datapplied Statisticspplied Statistics

:Population x:Sample

:Population :Sample2 s2

:Population :Sample s


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Mean

Mode

Median

easures of Central tendencyeasures of Central tendency: , , , ,Data 34 43 81 106 106 and

115

Average /x n = .0 83

Highest frequency = 106

( + )/Middle score 81 106 2 = .3 5


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:ariance

. , =Most of the data lies between 44 5 4 57 39 to 49

:tandard deviation

44

50

38

49

42

47

40

39

46

50

188.5

.4 5

- .5.5- .5.5- .5.5- .5- .5.5.5

.3.0 3.2 3.0 3.3.3.0 3.0 3.3.0 3

.0 9

4.57

x

2)( xx

=

n

i

ixx

1

2)( SS

/( - )S n 1 MS

sdMS

)( xx

easures of dispersioneasures of dispersion

x


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Coefficient of Variance = /V S* %100

. / . * % = . %4 57 44 5 100 10 28

. %Standard deviation is 10 28 of the mean



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GradeGrade ScoreScoreGenius 145Gifted 130-144Above average 115-129Higher average 100-114Lower average 85-99Below average 70-84Borderline low 55-69Low


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115 130 145100857055 145


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.4 13%.4 13%

.3 59%.3 59%

.14%.14%13% .13%

Pro

ba

bili

ty

Score

-6 -5 -4 -3 -2 -1 1 2 3 4 5Sdfrom

.0031% .000028%.0031%000028%

6

Normal Distribution


.8 2689% .5 4499% .9 7300% .9 9936%99.999942669% .9 999999802%


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-6 -5 -4 -3 -2 -1 1 2 3 4 5Sdfrom 6

Normal Distribution


.9 999999802%

0.0000001980.00198

USLLSL

DPMOPMOPHOPHO ixSigma


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easures of dispersioneasures of dispersionNormal Distribution

USLLSL USLLSL


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Normal Distribution


USLLSL

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

.1 5

.4 DMPO4 DMPO


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Normal Distribution


USLLSL

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

a

b

dc

= /Cp a b

= ( )/ .Cpk c or d 0 5b

Processcapability


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Non Normal Distribution


:Measurements

Kurtosisurtosis

Skewnesskewness

+ve-ve


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tatistical significance teststatistical significance testsSignificance

tests

--testest--testest--testest

ANOVANOVA


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+ :e z ,values are above the mean- :e z values are below the mean

1 point compared to population Group compared to population

Population

=

i

i

x

z

n

xz

=

tatistical significance teststatistical significance tests-Z

test-- aluealue:

How many tandard deviationsaway from mean?

s

xxz

=

Sample


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s

xxz

= 07.1

57.6

20.262.19=

=

.o this person has a BMI 1 07 standard deviations below the mean

hat is the probability that of a person having BMI.9 2 sd below the mean.9 2 sd above the mean


test

ean ( ) = .6 20(tandard deviation s) = .57x

ampleample:BMI

. :with a BMI of 19 2 has a z score of


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Pro

ba

bili

ty

Sd

-1

< .9 6 > .9 6

0

Standard deviation

Z score


testampleample:

-1

%4%6


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Test group : Employee having two wheelerTest : Commuting time from home to BioconClaim : Average commuting time is less than 24 min

Samples : 30

18 16 23 19 25 48 13 17 20 23

16 21 18 16 29 15 8 19 20 7

15 16 24 15 6 11 14 23 18 12

t .01 (evel of significance = . ):01s there enough evidence to support the research claim???


testPopulatioopulatio::


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testPopulatioopulatio::

:sumption Population is normally distributed

X24Mean

Pro

ba

bili

ty

Score


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ypothesistestingull hypothesis : H0

lternate hypothesis :H1

:omparison of means:omparison of means

H1 : x tcritical

Null hypothesis will be rejected

ttest tcritical

H0:H1:

21 xx =

21 xx

Rejectedejected1xSo is significantly different from2x

Plant height

tatistical significance teststatistical significance tests-t

testCase

1ffect of fertilizer on plant height

= -df 2n 2

30 19

25 27

35 31

21 7

14 19

46 0

28 34

40 22

16 25

30 12

32 15

40 12

31 16

25 26

35 29

35 14

25 2236 20

21 38


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Fertiliz

er

/w o

Fertilizer

x

303 181s2

t test =.1 8

t critical= .2 02

ttest Fcritical ( )at significant level

Rejectedejected

tatistical significance teststatistical significance tests-F

test

Ha:


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AN alysis O fVAriance

ne wayne way:

wo waywo way:

( )ffect of one factor variable

( )ffect of two factors variables ffect of interaction

tatistical significance teststatistical significance testsANOVA


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Strate:y

= MSbgMSwg

Compare variability w i t h i ngroup MSwg to b e t w e e ngroups MSbg

Between groups Within groups

Group 1 Group2 Group1 Group 2

tatistical significance teststatistical significance testsOne way ANOVA


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( ):actor Independent Variable ( , , )ay Mon Wed Sat( ):ffect Dependent Variable umber of attendees

re any effect of presentation day on number of attendees ?(ull hypothesis H0) : o effect (1= 2 = 3)(lternate hypothesis H1) : here is an effect (1 2

Is there any impact of day on number of attendees ?Is there any impact of day on number of attendees ?



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SSbg .48 44 x( ++)

55

60

51

65

72

65

55

72

68

60

75

67

75

65

80

75

67

68

77

83

67

56

65

83

67

53

65

49

54

61

65

72

63

64

54

65

63.75 71.75 61

65.5

Mon Wed Sat

77

14

163

2

68

2

77

68

18

14

127

11

11

46

68

11

23

14

28

127

23

248

46

127

36

64

16

144

49

0

16

121

4

9

49

16

638 768 524

.74 25

.3 06 .39 06 .20 25

SSbg /df.8 5

M W S

Number of Attendees

.3 06 .39 06 .20 25 =MSbg =

SSM SSW SSS

SSwg

+ +

= 1930SSwg /dfSwg =

SS

= =( = -f 3 1= ) ( = ( )-f 12x3 3= )3

2)( xx

x

3/x = 2)( xx



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=.74 25

.8 5 ==MSbgMSwgF ritical or

umerator degrees of freedom : 2enominator degrees of freedom : 3(t significance level ) : .05 = .17

Ftest >Fcritical

So there are enough evidence to reject nullhypothesis

% :At 95 confidence level we can say

That the variation between means is not justby chance

.40

H0: (All means are same no effect of Day) Rejectedejected

Day of presentation matterssignificantly



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( ):actors Independent Variable) :Gender

( ):ffect Dependent Variable ) Number of participantsative impact of gender or type of sprot?

(ull hypothesis H0a ) : o effect of gender

(lternate hypothesis H1) : here is an effect

) Type of sport

interaction between gender and type of sport?

(ull hypothesis H0b ) : o effect of type of sport(ull hypothesis H0c ) : o interaction

tatistical significance teststatistical significance testsTwo way ANOVA

Man Woman

Indoor Outdoor


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30, 40, 50 60, 70, 80

140, 150, 160 5, 10, 15

Man Woman

Indoor

Outdoor

Source Df SS MS F

Gender g-1 SSG MSG MSG/MswithinSports s-1 SSs MSs MSs /Mswithin

G x S (g-1)(s-1)

SSG x s MSG x

s

MSG x s/MSwithinWithin (k-1) x I

x jSSwithin MSwithi

nSource Df SS MS F Fcritical(=0.01)Gender 1 9075 9075 111.69 11.3

Sport 1 1875 1875 23.07 11.3

G x S 1 21675 21675 266.77 11.3

Within 8 650 81.3

gs



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Woman Man

Ind 70 50

Otd 10 150

Indoor Outdoor

(ull hypothesis H0a ) : o effect of gender Rejectedejected(ull hypothesis H0b ) : o effect of type of sports Rejectedejected

(ull hypothesis H0c ) : o interaction Rejectedejected



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30o C 35oC

30o C 35o C

pH7 70 50

pH5 10 150


H 5H 7

( ):actors Independent Variable :Temperature

( ):ffect Dependent Variable ) ( )Total product g

) pH30 35

5 7


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Investigation of relationship betweenvariables

X Y

2 48

19 30

34 17.5

40 11

8 41

12 42

20 35

20 31

37 18

19 35

30 16

46 8.3

egression and correlationegression and correlation:Regression analysis


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Investigation of relationship betweenvariables

X Y

2 48

19 30

34 17.5

40 11

8 41

12 42

20 35

20 31

37 18

19 35

30 16

46 8.3

=R.0 955

= - . +y 0 951x.50 49

=y ax+b

imple linear regressionimple linear regression

One independent variable



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y = ax + b

y = a1x1+ a2x2+ a3x3+ b

imple linearimple linearregressionegressionultiple linearultiple linearregressionegression

Linear Non Linear


ononlinearineary = a1x1+ a2x2+ a11x2 + a12x1x2+b


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Is the relationship we have describedstatistically significant?

-Significant tests

( )To find how well or badly a line fits theobservation

What is the strength of this relationship- r2 ( )coefficient of determination or djusted r2

egression and correlationegression and correlationCorrelation

:analysis


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= ax + bslope intercept

= , predicted value

= residual error =

= y i , true value

y -

( )A and b values are calculated that minimize Sum of Squares SS of residuals

(y )2 : minimum


:analysis


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Total Error

SSTotal

SSErrorr2 = 1


:analysis

SSTotal /(n-1)

SSError /(n-p-1)Adjustedr2 = 1

n= total observation

p= Number of predictor

(yi y)2 (y )2

r2 : oefficient of: oefficient ofdeterminationetermination

lways between 0 and 1ncrease with number of predictor

t can be negative alsorue representative of relationship st


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Group1

Group 2Group 1 Group 2

MSwg

MSbgF =

MSError

MSModelF =

Model Error


:analysistatistical significancetatistical significancef relationshipf relationship


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ne factor at time( )FATultiple factor at( )ime MFAT

esign of experimentesign of experimentTraditional method

Traditional method


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esign of experimentesign of experiment


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Number offactors

Screening Optimization Robustness

2-4 Full orfractional

factorial

Central composite

orBox-Behnken

Taguchi

5 or more Fraction factorial orPlackett Burman

Screen first toreduce factors

Taguchi

ow to select adesign?



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Continuousontinuous

Categoricalategorical

/Independent variable s

:Numeric any value between lower and upper value

. , ,eg Temperature pH concentration

/ - :Numeric non numeric only characters or levels. , , ,eg Gender operator type temperature

/Range of a factor s -1( )lower +1( )higher( )middle/ :Dependent variable s Response

/ain effect s/ain effect s / /Effect s due to individual factor s/nteraction effect s/nteraction effect s/Effect s due to interaction of multiple factor

When two or more effects can not be distinguished

.eg Main effect is confounded with interaction effects Main effects and interaction effects are aliased

-esign of experiment terminology-esign of experiment terminologyFactors

Levels

Effects

/Confounding Aliasing

esign of experiment


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Resolution type

Order ofinteraction effectsinteraction effectsconfounded with main effectmain effect

Experiment typeExperiment type

III 2 (eg. A with A.B or A.C or

B.C etc)

Screening

IV 3 (eg. A with ABC) OptimizationV 4 (eg A with ABCD) Optimization

rder interaction are less significant than lower order interaction

esign of experimentResolution of a design Power of a

design



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:ull factorial:ull factorial LfLevel

Factor

No. ofLevels

No. ofFactors

Designtype

Number ofexperiments2 2 22 2x2=4

2 3 23 2x2x2=8

3 2 32 3x3=9

3 3 3

3

3x3x3=27

Factorial

design



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22

4 experiments

Factorialdesign

a

b



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a

cb

8 experiments

23

Factorialdesign



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9 experiments

32

Factorialdesign

a

b



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27 experiments

33

Factorialdesign

cb



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23

8 experiments

2 -31

4 experiments

Fractional Factorialdesign



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Response surface methodology



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12 experiments

-oxBehnken

Geometry of some important response surface designs

.eg 3 factor 3 level



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entral compositedesign .eg 2 factor 2level

+ =




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aguchidesign:Inner array

:Outer array

Controllable variables during production

Uncontrollable variables during production

SignalNoise

, ,Media pH feed rate

, ,Temp DO


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