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In statistics, the **sign test** can be used to test the hypothesis that the difference median is zero between the continuous distributions of two random variables *X* and *Y*, in the situation when we can draw paired samples from *X* and *Y*. It is a non-parametric test which makes very few assumptions about the nature of the distributions under test - this means that it has very general applicability but may lack the statistical power of other tests such as the paired-samples t-test or the Wilcoxon signed-rank test.^{[citation needed]}

Let *p* = Pr(*X* > *Y*), and then test the null hypothesis H_{0}: *p* = 0.50. In other words, the null hypothesis states 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.

To test the null hypothesis, independent pairs of sample data are collected from the populations {(*x*_{1}, *y*_{1}), (*x*_{2}, *y*_{2}), . . ., (*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.^{[1]}

Then let *W* be the number of pairs for which *y*_{i} − *x*_{i} > 0. Assuming that H_{0} is true, then *W* follows a binomial distribution *W* ~ b(*m*, 0.5). The "*W*" is for Frank Wilcoxon who developed the test, then later, the more powerful Wilcoxon signed-rank test.^{[2]}

Let *Z*_{i} = *Y*_{i} – *X*_{i} for *i* = 1, ... , *n*.

- The differences
*Z*are assumed to be independent._{i} - Each
*Z*comes from the same continuous population._{i} - The values
*X*_{i}and*Y*_{i}represent are ordered (at least the ordinal scale), so the comparisons "greater than", "less than", and "equal to" are meaningful.

Since the test statistic is expected to follow a binomial distribution, the standard binomial test is used to calculate significance. The normal approximation to the binomial distribution can be used for large sample sizes, *m*>25.^{[1]}

The left-tail value is computed by Pr(*W* ≤ *w*), which is the p-value for the alternative H_{1}: *p* < 0.50. This alternative means that the *X* measurements tend to be higher.

The right-tail value is computed by Pr(*W* ≥ *w*), which is the p-value for the alternative H_{1}: *p* > 0.50. This alternative means that the *Y* measurements tend to be higher.

For a two-sided alternative H_{1} the p-value is twice the smaller tail-value.

- Wilcoxon signed-rank test - A more powerful variant of the sign test, but one which also assumes a symmetric distribution.
- Median test - An unpaired alternative to the sign test.

- ^
^{a}^{b}Mendenhall, W.; Wackerly, D. D. and Scheaffer, R. L. (1989), "15: Nonparametric statistics",*Mathematical statistics with applications*(Fourth ed.), PWS-Kent, pp. 674–679, ISBN 0-534-92026-8 **^**Karas, J. & Savage, I.R. (1967)*Publications of Frank Wilcoxon (1892–1965).*Biometrics 23(1): 1–10

- Gibbons, J.D. and Chakraborti, S. (1992). Nonparametric Statistical Inference. Marcel Dekker Inc., New York.
- Kitchens, L.J.(2003). Basic Statistics and Data Analysis. Duxbury.
- Conover, W. J. (1980). Practical Nonparametric Statistics, 2nd ed. Wiley, New York.
- Lehmann, E. L. (1975). Nonparametrics: Statistical Methods Based on Ranks. Holden and Day, San Francisco.