# 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*, 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.

Formally, 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.

## Method

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]}

## Significance testing

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.

## See also

## References

- ↑
^{1.0}^{1.1}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

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