Sign test

The sign test uses only the signs of independent differences $D_1,\dots ,D_n$. Under the Null hypothesis $\mathbb{P}(D_i>0)=\mathbb{P}(D_i<0)=1/2$ and no ties, the count $S={\sum }_i{1}_{\{D_i>0\}}$ has the binomial distribution $\mathrm{Binomial}(n,1/2)$. It can test a continuous population median against a specified value, or a median paired difference against zero, without specifying the magnitude distribution of the differences.

For a two-sided test, one conventional exact p-value is $\min \{1,2\min [\mathbb{P}(B\le S),\mathbb{P}(B\ge S)]\}$ for $B\sim \mathrm{Binomial}(n,1/2)$. Seven positive differences among eight give $2\mathbb{P}(B\ge 7)=18/256\simeq 0.0703$. The test is robust to the sizes of extreme differences because only signs are used, but discarding magnitudes can lose power relative to a suitable parametric test.

With zero differences, deleting ties gives an exact conditional sign calculation when the nonzero signs are equally likely under the Null hypothesis. For a discrete population, merely declaring zero a median does not by itself imply that conditional sign symmetry. Dependence between pairs also invalidates the independent binomial calculation.