What is the only meaningful measure of central location for a categorical variable?

The mode is the only meaningful measure of central location for a categorical variable because it identifies the most frequently occurring category within a dataset. Categorical variables represent distinct groups or categories that do not have a specific numerical value or order. Therefore, concepts such as mean and median, which rely on numerical calculations and order, are not applicable.

When analyzing categorical data, the mode gives insight into which category is most prevalent in the dataset, which can be crucial for understanding trends, preferences, or behaviors within the population being studied. For instance, if a survey includes responses about favorite colors, the mode would indicate the color chosen by the highest number of respondents.

In contrast, mean and median require a numerical context and cannot provide meaningful results for categorical data. Variance, being a measure of dispersion for quantitative variables, is also irrelevant for categorical variables as it requires numerical values to calculate how spread out the values are from the mean. Thus, mode stands out as the appropriate measure for summarizing categorical data.