ANOVA

Iqra University, Main Campus
Course: Statistical Inferences Faculty: Iftikhar Mubbashir
Date: December 5, 2013
Fall 2013

Statistics-Walpole Chapter-12

One way Classification
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Random samples of size n are selected from each of k populations.
The k populations are independent and normally distributed with means µ 1 , µ 2 ,K , µ k and common variance σ 2 .
We wish to derive appropriate methods for testing the hypothesis:

H 0 : µ1 = µ 2 = K = µ k Against, H 0 : at least two are unequal .
Let
x ij := The j th observation from the i th population.

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Ti := The total of all observations in the sample from the i th population.

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x i := The mean of all observations in the sample from the i th population.
T .. := The total of all nk observations.

Total

1 x 11 x 12
M
x 1n
T1

2 x 21 x 22
M
x 2n
T2

x1

xi

K
K
K K
K
K

Population i x i1 x i2
M
x in
Ti
xi

K
K
K K
K
K

k x k1 x k2
M
x kn
Tk

T .. x ..

xk

Sum of Squares Computational Formulas for equal sample sizes:
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k

n

2
TSS = ∑∑ x ij − i =1 j =1

k

T..2 nk

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∑T

2

i

T..2
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CSS = n nk
TSS = CSS + ESS i =1

Analysis of Variance for the One‐Way Classification for equal samples
Source of Variation
Column Means
Error

1

ESS

k (n − 1)

Total

Sum of Squares
CSS

Degree of Freedom

TSS

nk − 1

k −1

Mean Square
CSS
2 s1 = k −1
ESS
2 s2 = k (n − 1)

Computed f Value
2
s1 f= 2 s2 Sum of Squares Computational Formulas for equal sample sizes:
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k

n

2
TSS = ∑∑ x ij − i =1 j = 1

T..2 N

Ti 2 T..2
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N i =1 ni
• TSS = CSS + ESS
Source of Variation
Column Means

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