Statistics Knowledge Base - One-Way Analysis of Variance (ANOVA)

Wednesday, January 07, 2009

One-Way Analysis of Variance (ANOVA)

A one-way analysis of variance (ANOVA) is a statistical method through which the differences between the means of two or more independent groups can be evaluated. A one-way ANOVA is carried out by partitioning the total model sum of squares and degrees of freedom into a between-groups (treatment) component and a within-groups (error) component. The significances of inter-group differences in the one-way ANOVA are then evaluated by comparing the ratio of the between and within-groups mean squares to a Fisher F-distribution.

PROPERTIES

In order for a one-way analysis of variance to be tenable, several assumptions regarding the source data must be met. These assumptions include:

Homogeneity of variances - the variance of the data in each group must be statistically indistinguishable.
Case independence - the cases which comprise the dataset must be statistically independent within the context of probability theory.
Data normality - The data that comprise each group in the analysis must be normally-distributed.
Normality of error variances - the error variances for each group in the analysis must be normally (and independently) distributed.

ONLINE CALCULATOR

To calculate a one-way analysis of variance from summary data, please click here.

FORMULAE

The formulae involved in the computation of a one-way analysis of variance are detailed below.

ANOVA sums of squares formulae:
Where i is a score, n is the final score, j is a group, and p is the final group.

ANOVA mean squares formulae:

ANOVA F formula:

REFERENCES

Cohen, J., Cohen, P., West, S.G., and Aiken, L.S. (2003) "Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences (3rd edition)", Lawrence Earlbaum Associates, Mahwah, NJ

Ferguson, G.A., Takane, Y. (2005) "Statistical Analysis in Psychology and Education (6th edition)", McGraw-Hill, Montreal, Quebec