StatReason

Mood's Median Test

Compare medians across two or more independent groups using only whether each value falls above or at-or-below the overall median, the simplest and lowest-assumption test on this site for comparing central tendency across groups.

What this answers

This test answers "do these groups differ in where their values sit relative to the overall median?" using the least amount of information possible from each observation: not its rank, not its value, only whether it is above or at-or-below one shared threshold.

How it is calculated

First, the median across every group combined (the grand median) is computed. Then, for each group, this calculator counts how many values fall strictly above that grand median and how many fall at or below it, building a 2-row table with one column per group. A standard Pearson chi-square test, the same construction used for the Chi-Square Test of Independence, is then applied to that table.

Worked example

For three groups of 1, 2, 3, then 10, 11, 12, then 20, 21, 22, the grand median falls somewhere in the middle group's range, so the first group lands entirely at-or-below it, the third group lands entirely above it, and the middle group splits. The resulting chi-square statistic is large relative to 2 degrees of freedom, correctly detecting the obvious separation between groups even using only this coarse above/below information.

Assumption audit

Calculated from your data: the grand median, each group's above and at-or-below counts, and the resulting chi-square statistic and p-value.
Evidence to review: how many values sit exactly at the grand median; this calculator counts ties at the median as "at-or-below," a documented convention that can affect results when many values cluster exactly there.
You must verify: that your groups are independent of each other and that comparing medians (rather than means or full distributional shape) is genuinely the question you want answered.

What this result does not mean

A significant result means the groups' values are not evenly split above and below the grand median, not that the groups differ by a large amount; this test discards all information about how far above or below the median each value falls, which is exactly why it has less statistical power than a rank-based alternative on the same data.

Limitations

Because this test throws away so much information (only above/below matters, not magnitude or even relative rank), it typically has lower power than the Kruskal-Wallis Test to detect a real difference; use Mood's median test when you specifically want a test that depends only on medians and is minimally sensitive to shape or outliers elsewhere in each distribution.