How to use the sign test calculator
- Choose your data: a list of differences, two paired lists, or one list to compare with a median.
- Paste the values. For paired data, paste the second list in the same order. For a median test, enter the median to test against.
- Choose the alternative: two-sided, greater than or less than.
- Read the p value, the number of positive and negative differences and the number of zeros that were left out.
Formula
Drop the differences that equal 0, and count the positive (p) and negative (m) ones, with n = p + m
Under the null hypothesis each used difference is positive with probability 1/2, so p follows a binomial distribution with n trials
One-sided p value (greater) = P(X >= p) = sum of C(n, k) / 2^n for k from p to n
Two-sided p value = 2 x the smaller one-sided value, capped at 1
The test uses only the signs of the differences. The p value is exact, so there is no normal approximation.
Nine of ten differences are positive
With differences of 1.2, 0.8, 2.1, 0.5, 1.7, 3.0, 0.9, 1.1, -0.4 and 2.2, there are 9 positive and 1 negative. The two-sided p value is 2 x (1 + 10) / 1024 = 0.021484, which is below 0.05.
For paired readings of eight people before and after a change, with 7 higher and 1 lower, the two-sided p value is 0.070313. Testing 11 values against a median of 50 with 9 above and 2 below, in the direction greater than, gives 0.032715.
When to use a sign test
The sign test needs almost no assumptions: only that the differences are independent and come from a continuous distribution with a common median. That makes it a safe choice for skewed data, for ordinal scores and for small samples where a t test is hard to justify. The price is less power, because it ignores how big the differences are.
If you can rank the sizes of the differences, the Wilcoxon signed-rank test uses more of the information. For counts of successes directly, use the binomial probability calculator.
Ties and one-sided tests
A difference of exactly 0 gives no evidence either way, so it is dropped and the number of trials falls. If many values are zero, the test has little to work with. A one-sided test asks a more specific question, such as whether values are higher after a change, and you should choose the direction before looking at the data. A two-sided test asks only whether the median differs.
To turn another test statistic into a p value, use the p value calculator.
Assumptions and limits
- Pairs or observations are independent, and the differences come from a continuous distribution.
- Zero differences are dropped from the test.
- For paired data both lists have the same length and the same order.
- The p value is exact. It is only as meaningful as the design of the study.
Frequently asked questions
What does a sign test do?
It checks whether positive and negative differences are balanced. If one sign appears much more often than chance allows, the median is not zero.
How is the p value calculated?
From the binomial distribution with a probability of one half. For 9 positive out of 10 the two-sided value is 0.021484.
What happens to zero differences?
They are left out, and the page shows how many were dropped.
How is it different from the Wilcoxon test?
The sign test uses only the direction of each difference. The Wilcoxon test also uses their ranks, so it has more power when the sizes carry information.