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Problem: one way anova {{25,30,36,38,31},{31,39,38,42,35},{24,30,28,25,28}} [ Calculator, Method and examples ]

Solution:
Your problem `->` one way anova {{25,30,36,38,31},{31,39,38,42,35},{24,30,28,25,28}}


`A``B``C`
253124
303930
363828
384225
313528
`sum A=160``sum B=185``sum C=135`


`A^2``B^2``C^2`
625961576
9001521900
12961444784
14441764625
9611225784
`sum A^2=5226``sum B^2=6915``sum C^2=3669`


Data table
Group`A``B``C`Total
N`n_1=5``n_2=5``n_3=5``n=15`
`sum x_i``T_1=sum x_1=160``T_2=sum x_2=185``T_3=sum x_3=135``sum x=480`
`sum x_(i)^2``sum x_1^2=5226``sum x_2^2=6915``sum x_3^2=3669``sum x^2=15810`
Mean `bar x_i``bar x_1=32``bar x_2=37``bar x_3=27`Overall `bar x=32`
Std Dev `S_i``S_1=5.1478``S_2=4.1833``S_3=2.4495`


Let k = the number of different samples = 3
`n=n_1+n_2+n_3=5+5+5=15`

Overall `bar x=480/15=32`

`sum x=T_1+T_2+T_3=160+185+135=480 ->(1)`

`(sum x)^2/n=480^2/15=15360 ->(2)`

`sum T_i^2/n_i=(160^2/5+185^2/5+135^2/5)=15610 ->(3)`

`sum x^2=sum x_(1)^2+sum x_(2)^2+sum x_(3)^2=5226+6915+3669=15810 ->(4)`



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

`=15610-15360`

`=250`

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

`=5xx(32-32)^2+5xx(37-32)^2+5xx(27-32)^2`

`=250`

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

`=15810-15610`

`=200`

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






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