Friedman's Two-Way Analysis of Variance by Ranks is a nonparametric test designed to identify differences across multiple test attempts when traditional assumptions of normality and equal variances do not apply. Unlike conventional ANOVA, which requires normally distributed data with equal variances, Friedman's test is ideal for ordinal or non-normally distributed data, making it particularly useful for analyzing dependent samples, such as matched subjects over time or repeated measures from the same group.

The null hypothesis (H₀) in Friedman's test states that there is no difference in the distributions of the variables being compared. Rejecting H₀ indicates significant differences, not just in the central tendency (medians) but also in the shape and spread of the distributions. The process begins by ranking the data within each subject across the different conditions. The sum of ranks for each condition is then calculated, followed by the computation of Friedman's F, which assesses the significance of the differences. For larger samples, the test statistic is compared against critical values from either Friedman's distribution or the Chi-Square distribution.

For example, imagine a study evaluating the effectiveness of three different teaching methods on the test scores of the same group of students. After applying each method and recording the scores, the data are ranked within each student for the three methods. The sum of ranks for each method is then used to calculate Friedman's F. If the calculated F exceeds the critical value, it suggests that at least one of the teaching methods leads to significantly different scores, prompting further investigation.

Friedman's test provides a robust alternative when data do not meet the strict assumptions of parametric tests, ensuring researchers can still draw meaningful insights even when working with non-normal or ordinal data. Its versatility and flexibility make it a valuable tool for a wide range of research fields and datasets.