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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 25 31 24 30 39 30 36 38 28 38 42 25 31 35 28 sum A=160 sum B=185 sum C=135

 A^2 B^2 C^2 625 961 576 900 1521 900 1296 1444 784 1444 1764 625 961 1225 784 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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