The sign test is a nonparametric method used to evaluate hypotheses about the median of a single sample or to compare the medians of two related samples. The sign test is particularly useful when dealing with nominal data, which includes distinct categories without an inherent order, such as names, labels, and preferences. Nominal data restricts statistical analysis to evaluating population proportions rather than mean or median values that require continuous data.

For example, consider a survey that queries individuals about their pet preferences, yielding results where a certain number prefer dogs over cats. This scenario presents nominal data because pet preferences are categorical and cannot be ranked. The sign test can then be applied to determine if there's a statistically significant preference for dogs over cats (or the opposite) among the surveyed population.

In this example, the procedure involves two hypotheses: the null hypothesis (H₀), which posits no preference between dogs or cats (assuming an equal proportion of preferences), and the alternative hypothesis (H₁), suggesting a significant preference for one over the other. The test uses positive and negative signs to represent preferences for each category. The calculation of the test statistic, often transformed into a z-score for large samples (n > 25), is used to determine if the observed distribution of preferences significantly deviates from what would be expected under the null hypothesis.

If the computed statistic crosses a critical value at a chosen confidence level (e.g., 0.05), the null hypothesis is rejected, indicating no significant preference within the population. This method provides a straightforward approach to testing categorical data for differences, offering insights into population preferences without requiring assumptions about the data distribution.