Sign test

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In statistics, the sign test can be used to test the hypothesis that there is "no difference" between the continuous distributions of two random variables X and Y. Formally:

Let p = Pr(X > Y), and then test the null hypothesis H₀: p = 0.50. This hypothesis implies that given a random pair of measurements (x_i, y_i), then x_i and y_i are equally likely to be larger than the other.

Independent pairs of sample data are collected from the populations {(x₁, y₁), (x₂, y₂), . . ., (x_n, y_n)}. Pairs are omitted for which there is no difference so that there is a possibility of a reduced sample of m pairs.

Then let w be the number of pairs for which y_i − x_i > 0. Assuming that H₀ is true, then W follows a binomial distribution W ~ b(m, 0.5).

The left-tail value is computed by Pr(W ≤ w), which is the p-value for the alternative H₁: p < 0.50. This alternative means that the X measurements tend to be higher.

The right-tail value is computed by Pr(B ≥ w), which is the p-value for the alternative H₁: p > 0.50. This alternative means that the Y measurements tend to be higher.

For a two-sided alternative H₁ the p-value is twice the smaller tail-value.