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Statistics and DOE
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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
easures of dispersioneasures of dispersion
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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
easures of dispersioneasures of dispersion
.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
easures of dispersioneasures of dispersion
.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
easures of dispersioneasures of dispersion
USLLSL
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
.1 5
.4 DMPO4 DMPO
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Normal Distribution
easures of dispersioneasures of dispersion
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
easures of dispersioneasures of dispersion
: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
tatistical significance teststatistical significance tests-Z
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
tatistical significance teststatistical significance tests-Z
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???
tatistical significance teststatistical significance tests-Z
testPopulatioopulatio::
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tatistical significance teststatistical significance tests-Z
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 ?
tatistical significance teststatistical significance testsOne way ANOVA
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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
tatistical significance teststatistical significance testsOne way ANOVA
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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
tatistical significance teststatistical significance testsOne way ANOVA
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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
tatistical significance teststatistical significance testsTwo way ANOVA
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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
tatistical significance teststatistical significance testsTwo way ANOVA
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30o C 35oC
30o C 35o C
pH7 70 50
pH5 10 150
tatistical significance teststatistical significance testsTwo way ANOVA
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
egression and correlationegression and correlation:Regression analysis
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y = ax + b
y = a1x1+ a2x2+ a3x3+ b
imple linearimple linearregressionegressionultiple linearultiple linearregressionegression
Linear Non Linear
egression and correlationegression and correlation:Regression analysis
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
egression and correlationegression and correlationCorrelation
:analysis
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Total Error
SSTotal
SSErrorr2 = 1
egression and correlationegression and correlationCorrelation
: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
egression and correlationegression and correlationCorrelation
: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?
esign of experimentesign of experiment
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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
esign of experimentesign of experiment
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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
esign of experimentesign of experiment
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22
4 experiments
Factorialdesign
a
b
esign of experimentesign of experiment
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a
cb
8 experiments
23
Factorialdesign
esign of experimentesign of experiment
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9 experiments
32
Factorialdesign
a
b
esign of experimentesign of experiment
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27 experiments
33
Factorialdesign
cb
esign of experimentesign of experiment
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23
8 experiments
2 -31
4 experiments
Fractional Factorialdesign
esign of experimentesign of experiment
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Response surface methodology
esign of experimentesign of experiment
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12 experiments
-oxBehnken
Geometry of some important response surface designs
.eg 3 factor 3 level
esign of experimentesign of experiment
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entral compositedesign .eg 2 factor 2level
+ =
Geometry of some important response surface designs
esign of experimentesign of experiment
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aguchidesign:Inner array
:Outer array
Controllable variables during production
Uncontrollable variables during production
SignalNoise
, ,Media pH feed rate
, ,Temp DO
Geometry of some important response surface designs