ANOVA test the hypotheses that:
H0: 
;
Ha: Not all the means are equal |
The within-sample or treatment
variance
or variation is the average of the all the variances for each population
and is an estimate of 
whether the null hypothesis, H0 is true or not. The within-sample
variance is often called the unexplained variation.
The between-sample variance
or error is the average of the square variations of each population mean
from the mean
or all the data (Grand Mean, 
)
and is a estimate of 
only if the null hypothesis, H0 is true. The between-sample
variance is associated with the explained variation of our experiment.
The F-Distribution is the ratio
of the between-sample estimate of 
and the within-sample estimate:
The sum of squares for the between-sample variation is either
given by the symbol SSB (sum of squares between)
or SSTR (sum of squares for treatments) and is the explained
variation.
To calculate SSB or SSTR, we sum the squared deviations of the
sample treatment means from the grand mean
and multiply by the number of observations for each sample.
The sum of squares for the within-sample variation is either
given by the symbol SSW (sum of square within)
or SSE (sum of square for error).
To calculate the SSW we first obtained the sum of squares for each sample and then sum them.
The Total Sum of Squares, SSTO = SSB + SSW
The between-sample variance, 
,
where k is the number of samples or treatment and is often
called the Mean Square Between, 
,
The within-sample variance, 
,
where n is the total number of observations in all the samples is
often
called the Mean Square Within or Mean Square Error, 
,
Acceptance Criteria for ANOVA
| That is accept H0 if: F-Statistics < F-table or P-value > alpha. |