In ANOVA logic, partitioning the sources of variance and hypothesis testing can be done individually. According to the ANOVA method, the total variation is decomposed into two parts: a source of variation due to the group or factor effect (expressed with
) and a source of variation due to the measurement error (expressed with
This Demonstration uses the ANOVA table algorithm based on deviations.
By definition, the variance
is the sum of squared deviations;
is the total deviation, based on the differences between variants
and the mean of statistical complex;
is the intergroup or between-group deviation, based on differences between each group (or sample) and the complex means; and
is the intragroup or within-group deviation, based on differences between the variants
and the sample (group) mean.
This Demonstration illustrates some basic principles of one-way ANOVA only. We know that the Fisher
-test is used for comparisons of the components of the total deviation. The
-value is the ratio of variance between and variance within samples (groups).
Consider an experiment to study the effect of three different levels of a factor on a response
(here we use
but say there are
groups) observations for each level, we write the outcome of the experiment in a work table.
To calculate the
Calculate the size of the statistical complex (first row in the work table)
Calculate the sum
within each sample (group), and the complex total sum
, adding values in the row.
, the square of the sum.
Find the value
and the total sum in the row:
(sum of square).
Find the total sum in the row
Find the value
Calculate total variance
, between-group variance
, and within-group variance
The statistical ANOVA complex degrees of freedom is
The between-group degrees of freedom is one less than the number of groups (samples):
The within-group degrees of freedom is
And finally we find the
and the relative factor effect:
It is well known that the
-test is used to compare the components of the total deviation. In this Demonstration, we are using the textbook method of concluding the hypothesis test: compare the observed
-value with the critical value of
determined from tables. The critical value of
is a function of the numerator degrees of freedom, the denominator degrees of freedom, and the significance level alpha. If the experimental
-value is more than critical
-value, then reject the null hypothesis.
The Demonstration has thus illustrated how we can get and apply the experimental
 A. Gelman, "Analysis of Variance—Why It Is More Important Than Ever," Annals of Statistics
(1), 2005. pp. 1–53. doi:10.1214/009053604000001048
 D. C. Montgomery, Design and Analysis of Experiments
, 5th ed., New York: Wiley, 2001.