A measure of variation for categorical data is discussed. We develop an analysis of variance for a one-way table, where the response variable is categorical. The data can be viewed alternatively as falling in a two-dimensional contingency table with one margin fixed. Components of variation are derived, and their properties are investigated under a common multinomial model. Using these components, we propose a measure of the variation in the response variable explained by the grouping variable. A test statistic is constructed on the basis of these properties, and its asymptotic behavior under the null hypothesis of independence is studied. Empirical sampling results confirming the asymptotic behavior and investigating power are included.