## The sign test

Copyright © Richard B. Darlington. All rights reserved.
The sign test is used to test the null hypothesis that two groups are of equal size--for instance
the hypothesis that equal numbers of people respond Yes and No to a questionnaire.
It can also be used to test a hypothesis about a median, because the hypothesis that
a median equals, say, 40, is the hypothesis that equal numbers of cases fall above and below
40. To perform the test, simply count the numbers of cases above and below the
hypothesized median (ignoring any cases that exactly equal the hypothesized median), and
then perform the test.

The sign test is actually the binomial test with the null proportion set to .5. You
might use the binomial test, say, to test the null hypothesis that a 5-choice multiple-choice
test item is gotten right by exactly 20% of the people answering it; that is the proportion
expected by chance if everyone guesses. The only reason we have the separate name "sign
test" is that there are tables and approximate formulas that apply only to the case where the
null proportion is .5, and not to the more general binomial case.

There are at least 5 separate ways to execute the sign test; some are exact and some
approximate. The five ways below are listed approximately in order of increasing
difficulty:

- Use a sign-test table
(The one here is quite large.)

- Use the Pearson-Yates normal
approximation

- Use a much better normal approximation based
on the
likelihood-ratio test ##

- Work out exact numbers of
combinations

- Use a program for the binomial
distribution
##

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