This is a hypothesis testing procedure used to evaluate mean differences between two or more treatments (or populations) giving it an advantage over using t statistics which are limited to situations that only have two treatments to compare.

The hypotheses in this testing method are essentially the same as in other hypothesis tests:

1) The null hypothesis states that there are no significant differences between populations or treatments (H_{0} = μ_{1} = μ_{2} = μ_{3}).

2) The alternate hypothesis states that the populations or treatments really do have different means that are responsible for causing systematic differences between the sample means. In other words, it says that there is a treatment effect. There are many different possibilities for an alternative hypothesis. (H_{1} = μ_{1} = μ_{3} but not μ_{2} -or- H_{1} = μ_{1} ≠ μ_{2} ≠ μ_{3} are some of the possibilities)

The test statistic for this hypothesis testing procedure is called the F-ratio. This has the same basic structure as the t-statistic except that it is based on variance as opposed to mean difference. It is defined by the variance (differences) between sample means divided by the variance expected with no treatment. The larger the value that the F-ratio is, the more evidence that the differences in the sample mean are larger than would be predicted if there were no treatment effect.

Notations:

1) K is used to identify the number of treatment conditions and in an independent-measures study also denotes the number of samples (ex. k = 3 means there are three treatments or samples).

2) Sample size is still denoted by n using subscripts if the individual samples have different sample sizes.

3) The total number of scores (individuals) in the entire study is denoted by N. When n is a constant than N = kn.

4) The sum of the scores, ∑X, is denoted with T for treatment total. The total for a specific treatment is identified by a subscript.

5) The sum of all scores in the study is denoted with G where you can add up all scores, N, or add up the treatment totals, T (G = ∑T).

6) Remember SS has to be calculated differently for the different variances. (if k = 3, than it requires that you find 3 values for SS and df, two variances, and the F-ratio).

Hypothesis testing using this procedure:

1) State the hypothesis and select the alpha level

2) Locate the critical region by determining the df for both MS_{between} and MS_{within} and using the F distribution table to locate the alpha level that defines the critical region.

3) Compute the F-ratio by obtaining the SS, using those values and the df values to compute the two MS values, and using those values to calculate the F-raito.

4) Determine whether to reject or confirm the H_{0} by seeing if the obtained F value is within the critical region.

Assumptions made in an independent-measures ANOVA hypothesis test:

1) The observations within each sample are independent of one another.

2) Populations that the samples are derived from must be normal.

3) Homogeneity of variance across populations.