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1. One-way ANOVA method example ( Enter your problem )
  1. Example-1
  2. Example-2
Other related methods
  1. One-way ANOVA method
  2. Two-way ANOVA method

1. Example-1


1. Solve using One-way ANOVA method
ObservationABCDE
18101287
212119144
318121668
4139121615


Solution:
`A``B``C``D`
8121813
1011129
1291612
814616
74815
`sum A=45``sum B=50``sum C=60``sum D=65`


`A^2``B^2``C^2``D^2`
64144324169
10012114481
14481256144
6419636256
491664225
`sum A^2=421``sum B^2=558``sum C^2=824``sum D^2=875`


Data table
Group`A``B``C``D`Total
N`n_1=5``n_2=5``n_3=5``n_4=5``n=20`
`sum x_i``T_1=sum x_1=45``T_2=sum x_2=50``T_3=sum x_3=60``T_4=sum x_4=65``sum x=220`
`sum x_(i)^2``sum x_1^2=421``sum x_2^2=558``sum x_3^2=824``sum x_4^2=875``sum x^2=2678`
Mean `bar x_i``bar x_1=9``bar x_2=10``bar x_3=12``bar x_4=13`Overall `bar x=11`
Std Dev `S_i``S_1=2``S_2=3.8079``S_3=5.099``S_4=2.7386`


Let k = the number of different samples = 4
`n=n_1+n_2+n_3+n_4=5+5+5+5=20`

Overall `bar x=220/20=11`

`sum x=T_1+T_2+T_3+T_4=45+50+60+65=220 ->(1)`

`(sum x)^2/n=220^2/20=2420 ->(2)`

`sum T_i^2/n_i=(45^2/5+50^2/5+60^2/5+65^2/5)=2470 ->(3)`

`sum x^2=sum x_(1)^2+sum x_(2)^2+sum x_(3)^2+sum x_(4)^2=421+558+824+875=2678 ->(4)`



ANOVA:
Step-1 : sum of squares between samples
`"SSB"= (sum T_i^2/n_i) - (sum x)^2/n = (3)-(2)`

`=2470-2420`

`=50`

Or
`"SSB"=sum n_j * (bar x_j - bar x)^2`

`=5xx(9-11)^2+5xx(10-11)^2+5xx(12-11)^2+5xx(13-11)^2`

`=50`

Step-2 : sum of squares within samples
`"SSW"= sum x^2 - (sum T_i^2/n_i) = (4)-(3)`

`=2678-2470`

`=208`

Step-3 : Total sum of squares
`"SST"="SSB"+"SSW"`

`=50+208`

`=258`

Step-4 : variance between samples
`"MSB"=("SSB")/(k-1)`

`=50/(3)`

`=16.6667`

Step-5 : variance within samples
`"MSW"=("SSW")/(n-k)`

`=208/(20-4)`

`=208/(16)`

`=13`

Step-6 : test statistic F for one way ANOVA test
`F=("MSB")/("MSW")`

`=16.6667/(13)`

`=1.2821`

the degree of freedom between samples
`k-1=3`

Now, degree of freedom within samples
`n-k=20-4=16`

ANOVA table
Source of VariationSums of Squares
SS
Degrees of freedom
DF
Mean Squares
MS
F
Between samplesSSB = 50`k-1` = 3MSB = 16.66671.2821
Within samplesSSW = 208`n-k` = 16MSW = 13
TotalSST = 258`n-1` = 19



This material is intended as a summary. Use your textbook for detail explanation.
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