Applied Statistics And Doe Mayank

Applied Statistics and DOE
Mayank

Applied Statistics
Measures of central tendency (central position of data)
µ
Mean
Population :
Sample:
Median
Mode
Measures of dispersion (spread of data)
Variance
σ2
s2
Population :
Sample:
Standard deviation
σ
s
Population :
Sample:
Coefficient of variation

Measures of Central tendency
Data: 34, 43, 81, 106, 106 and 115
Mean
Average Σx/n =80.83
Mode
Highest frequency =106
Median
Middle score (81+106)/2 =93.5

Measures of dispersion
Variance:
Standard deviation:
x
SS
SS/(n-1)
MS
sd
√MS
Most of the data lies between 44.5±4,57 = 39to 49

Coefficient of Variance
CV = s/ *100%
4.57/44.5*100% = 10.28%
Standard deviation is 10.28% of the mean

Normal Distribution
Example: IQ Score

Normal Distribution
IQ Score
Count
Score
<55
115
130
145
100
85
70
55
145<

Normal Distribution
34.13%
34.13%
Probability
13.59%
13.59 %
Score
2.14%
2.14%
0.13%
0.13%
0.0031%
0.0031%
0.000028%
0.000028%
Sd from
-6σ
-5σ
-4σ
-2σ
-1σ
1σ
2σ
3σ
-3σ
5σ
6σ
4σ
μ
68.2689%
95.4499%
99.7300%
99.9936%
99.999942669%
99.999999802%

Normal Distribution
Six Sigma
DPMO
DPHO
LSL
USL
Sd from
-6σ
-5σ
-4σ
-2σ
-1σ
1σ
2σ
3σ
-3σ
5σ
6σ
4σ
μ
99.999999802%

Normal Distribution
LSL
LSL
USL
USL

Normal Distribution
1.5 σ
LSL
USL
3.4 DMPO
-6σ
-5σ
-4σ
-2σ
-1σ
1σ
2σ
3σ
4σ
-3σ
5σ
6σ
μ

Statistical significance tests
Significance tests
Z- test
t- test
F- test
ANOVA

Z - test
Z-value :
How many standard deviations away from mean?
+ve z: values are above the mean,
-ve z: values are below the mean
Population
Sample
Group compared to population
1 point compared to population

Z - test
Sample :
BMI
Mean ( ) = 26.20
Standard deviation (s) = 6.57
What is the probability that of a person having BMI
19.2 sdbelow the mean
19.2 sd above the mean
A person with a BMI of 19.2 has a z score of:
So this person has a BMI 1.07 standard deviations below the mean

Z - test
Sample :
Probability
<19.6
>19.6
Sd
16 %
84 %
-1σ
μ
Standard deviation
Z score
0
-1

Z - test
Population :
Test group : Employee having two wheeler
Test : Commuting time from home to Biocon
Claim : Average commuting time is less than 24 min
At 0.01 level of significance (α=0.01):
Is there enough evidence to support the research claim???
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

Z - test
Population :
Assumption: Population is normally distributed
Probability
Score
24
Mean
X

Z - test
Population :
Hypothesis testing
Test vs Population
Comparison of means:
Null hypothesis : H0
No difference (Claim not true)
H0 : x ≥ µ
µ = 24
Alternate hypothesis : H1
It is different (Claim is true)
H1 : x < µ

Z - test
Population :
Probability
Probability
24
Mean
X
Z value
Score
Level of significance
α = 0.01
Critical
value
Z
0
-2.33

Z - test
Population :
Ztest< Zcritical
Ztest>Zcritical
Rejection region
Acceptance region
-2.33
Z
= 18.2
s = 7.7
Z = - 4.13
µ = 24
n = 30

Z - test
Population :
Rejection region
-2.33
- 4.13
Z
So is test value is significantly different (lower) than the mean
Yes: There are significant evidence to reject the null hypothesis
H0 : s ≥ 24
Rejected
and therefore accept the claim
H1 : s < 24
Significantly supported

t - test
Comparison of means between two groups
H0:
H1:
Null hypothesis will be rejected
ttest > tcritical
Null hypothesis will not be rejected
ttest < tcritical

t - test
Comparison of means between two groups
Signal
Difference between group means
t =
=
Noise
Variability of groups

t - test
Effect of fertilizer on plant height
Case 1
Fertilizer
w/o Fertilizer
27.15 – 17.9
t test =
= 2.4
t critical with 38 df
at 0.05 significance level
= 2.03
Plant height
df = 2n-2
ttest > tcritical
So is significantly different from
H0:
Rejected
H1:
s2

t - test
Case 2
Fertilizer
w/o Fertilizer
t critical =2.03
1.3
t test =
Plant height
ttest < tcritical
So is not significantly different from
H0:
Not rejected
Rejected
H1:
s2

t - test
Overview

F - test
Comparison of variances
where and are the sample variances
F =
The F hypothesis test is defined as:
H0: =
Rejected
Ha:
<
>
≠
If Ftest > Fcritical (at significant level)

ANOVA
ANalysisOf VAriance
One way :
Effect of one factor (variable)
Two way :
Effect of two factors (variables)
Effect of interaction

One way ANOVA
Strategy:
Compare variability within group MSwg to between groups MSbg
MSbg
F =
MSwg
Group 1
Group 1
Group 2
Group 2
Between groups
Within groups

One way ANOVA
Is there any impact of exam room temperature on student performance?
Factor ( Independent Variable):
Temperature (cold, optimum, hot)
Effect ( Dependent Variable):
Score (marks obtained)
Null hypothesis (H0) : No effect (µ1= µ2 = µ3)
Alternate hypothesis (H1) : There is an effect (µ1 ≠ µ2 ≠ µ3)

One way ANOVA
C
O
H
Cold
Opt
Hot
Number of Attendees
SS
= X̄

One way ANOVA
MSbg
=
=
F =
6.40
MSwg
Fcriticalfor
Numerator degrees of freedom : 2
Denominator degrees of freedom : 33
At significance level (α) : 0.05
=
4.17
Ftest > Fcritical
So there are enough evidence to reject null hypothesis
H0: All means are same (no effect of Temperature)
Rejected
At 95% confidence level we can say:
That the variation between means is not just by chance
Examination Room temperature matters significantly

Two way ANOVA
Factors ( Independent Variable):
1) Gender:
Man Woman
2) Type of sport
Indoor Outdoor
1) Number of participants
Relative impact of gender or type of sprot?
Any interaction between gender and type of sport?
Null hypothesis (H0a) : No effect of gender
Null hypothesis (H0b) : No effect of type of sport
Null hypothesis (H0c) : No interaction
Alternate hypothesis (H1) : There is an effect

Two way ANOVA
Man Woman
s
↓
g->
Indoor
Outdoor

Two way ANOVA
Indoor Outdoor
Null hypothesis (H0a) : No effect of gender
Rejected
Rejected
Null hypothesis (H0b) : No effect of type of sports
Rejected
Null hypothesis (H0c) : No interaction

Two way ANOVA
Factors ( Independent Variable):
1) Temperature:
30 35
2) pH
5 7
1) Total product (g)
pH 7
pH 5
30o C 35o C

Regression and correlation
Regression analysis:
Investigation of relationship between variables

Investigation of relationship between variables
y = -0.951x + 50.49
y = ax +b
R² = 0.955
One independent variable
Simple linear regression

Simple linear regression
y = ax + b
Non linear
Multiple linear regression
y = a1x1+ a2x2+ a11 x2 + a12 x1x2+b
y = a1x1+ a2x2+ a3x3+ b
Linear
Non Linear

Correlation analysis:
To find how well (or badly) a line fits the observation
What is the strength of this relationship
- r2 (coefficient of determination) or adjusted r2
Is the relationship we have described statistically significant?
-Significant tests

ŷ = ax + b
intercept
slope
ε
= ŷ, predicted value
= y i, true value
ε =residual error =
y - ŷ
A and b values are calculated that minimize Sum of Squares (SS) of residuals =
Σ (y – ŷ)2 : minimum

r2 : Coefficient of determination
Error
Total
(yi – y)2
(y – ŷ)2
Always between 0 and 1
Increase with number of predictor
SSError
= 1-
r2
SSTotal
It can be negative also
SSError/(n-p-1)
Adjusted r2
= 1-
True representative of relationship strength
SSTotal/(n-1)
n= total observation
p= Number of predictor

MSbg
MSModel
=
=
F
F
MSwg
MSError
Group 1
Group 1
Group 2
Group 2
Statistical significance of relationship
Error
Model

Design of experiment
Traditional method
One factor at time (OFAT)
Statistical method
Multiple factor at time (MFAT)

How to select a design?

Design of experiment- terminology
Independent variable/s
Factors
Continuous
Numeric: any value between lower and upper value
eg. Temperature, pH, concentration
Categorical
Numeric/non-numeric : only characters or levels
eg. Gender, operator, type, temperature
Levels
-1(lower)
+1(higher)
0(middle)
Range of a factor/s
Effects
Dependent variable/s: Response
Main effect/s
Effect/s due to individual factor/s
Interaction effect/s
Effect/s due to interaction of multiple factors
Confounding/Aliasing
When two or more effects can not be distinguished
eg. Main effect is confounded with interaction effects
Main effects and interaction effects are aliased

Resolution of a design
Power of a design
Higher order interaction are less significant than lower order interaction

Factorial design
Factor
Lf
Full factorial:
Level

Factorial design
22
b
a
4 experiments

Factorial design
23
c
b
a
8 experiments

Factorial design
32
b
a
9 experiments

Factorial design
33
c
b
27 experiments

Fractional Factorial design
23-1
23
8 experiments
4 experiments

Response surface methodology

Geometry of some important response surface designs
Box - Behnken
eg. 3 factor 3 level
12 experiments

Central composite design
eg. 2 factor 2level
+
=

Taguchi design
Signal
Media, pH, feed rate
Inner array:
Controllable variables during production
Outer array:
Uncontrollable variables during production
Noise
Temp, DO,

Applied Statistics And Doe Mayank

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