7.2 The Median Test

The median test compares \(k\) samples by asking a single question of each observation: is it above the combined median, or not?

Pool all \(N\) observations and find their median \(M\). For each sample count how many observations exceed \(M\) and how many do not, giving a \(2 \times k\) contingency table. Under \(H_0\) that all \(k\) samples come from populations with the same median, each sample should split in the same proportion, and the table is tested with the usual statistic \[\chi ^{2} = \sum \dfrac {(O - E)^{2}}{E}, \hspace {0.8cm} E_{ij} = \dfrac {(\text {row total})(\text {column total})}{N},\] on \((k-1)\) degrees of freedom. For \(k = 2\) and small samples, Fisher’s exact test replaces the approximation.

Its virtue is that it needs almost nothing of the data — only a comparison against one number — so it survives censoring, coarse recording, and ordinal scales. Its vice is the same fact: discarding everything except above-or-below throws away most of the information, and the median test is markedly less powerful than Kruskal–Wallis when both apply. Prefer Kruskal–Wallis unless the data will not support it.

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