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

, where

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

,

,

. With

(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

-ratio:

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.

Calculate

, the square of the sum.

Find the value

and the total sum in the row:

.

Calculate the

(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

-ratio:

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

-value.

[1] A. Gelman, "Analysis of Variance—Why It Is More Important Than Ever,"

* Annals of Statistics*,

**33**(1), 2005. pp. 1–53. doi:

10.1214/009053604000001048.

[2] D. C. Montgomery,

*Design and Analysis of Experiments*, 5th ed., New York: Wiley, 2001.