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Analysis of Variance (ANOVA) is a statistical technique for detecting differences among the means of groups within a sample. It is one of several techniques of the ”general linear model.”
In the basic case, a sample is divided into groups based on values of one discrete independent variable with a small number of categories. Within each group, the means for a second variable, the dependent variable, are calculated. The difference in the means for the different groups is compared to the variation of the individual cases within each group around that group’s mean. The larger the difference in the means (relative to the variation around each mean), the more likely it is that the means are significantly different, and the less likely that one would make a type I (alpha) error by saying that the groups have different means in the population from which the sample is drawn.
Key to ANOVA, the F statistic comprises the ratio of the mean squared error between groups and the mean squared error within groups. The larger the difference between means of each group, the larger the F ratio is (holding constant the variation around the individual means). The larger the variation around each individual mean, the smaller is the F ratio (holding constant the difference between the means for each group). To make reliable inferences about the population based on the sample, ANOVA assumes: the sample was drawn randomly from the population, and the distribution of the dependent variable around the mean(s) is normal, not skewed in either direction.
F ratios are distributed in a family of curves based on the degrees of freedom for the between group means (number of groups of the independent variable minus one) and the degrees of freedom within groups (number of individual cases minus the number of values of the independent variable).
- Agresti, A. & Finlay, B. (1997). Statistical Methods for the Social Sciences, 3rd edn. Prentice Hall, Upper Saddle River, NJ.