Sign Test Calculator

The Sign Test is a non-parametric hypothesis test used to assess claims about a population median. It is ideal for paired samples, ordinal data, or very small samples when normality cannot be assumed. Use it when the Wilcoxon Signed-Rank Test is too complex or when your data only supports a direction of difference (positive or negative), not the magnitude.

The Sign Test counts the number of positive differences and negative differences between paired observations, ignoring ties (zero differences). The test statistic S is the smaller of the two counts (for a two-tailed test). It then uses the binomial distribution with p = 0.5 to compute the probability of observing that many or fewer successes under the null hypothesis, yielding a p-value.

Ties occur when two paired values are equal, resulting in a difference of zero. The Sign Test cannot assign a direction (positive or negative) to a tie, so ties are excluded from the effective sample size n. Only the non-zero differences are used in the calculation.

Choose a two-tailed test (H₁: M ≠ 0) when you simply want to detect any difference in the median without a pre-specified direction. Choose a right-tailed test (H₁: M > 0) if you expect the first group to have a higher median, or a left-tailed test (H₁: M < 0) if you expect a lower median. The alternative hypothesis should be decided before collecting data.

For a two-tailed test, the test statistic S is the minimum of the positive count and the negative count (S = min(n+, n−)). For a right-tailed test, S = n− (the number of negatives), and for a left-tailed test, S = n+ (the number of positives). The p-value is then the cumulative binomial probability P(X ≤ S) under n trials and p = 0.5, multiplied by 2 for a two-tailed test.

The significance level α (commonly 0.05) is the threshold for rejecting the null hypothesis. If the computed p-value is less than or equal to α, you reject H₀ and conclude the result is statistically significant. If p > α, you fail to reject H₀. A smaller α (e.g., 0.01) makes the test more stringent and reduces the chance of a false positive.

The Wilcoxon Signed-Rank Test considers both the direction and the magnitude (rank) of differences, making it more powerful than the Sign Test when the magnitude information is available and reliable. The Sign Test only uses the direction (positive or negative), making it simpler and applicable when only ordinal comparisons are possible or sample sizes are very small.

Yes. For a one-sample Sign Test, you compare each observation to a hypothesized median value. Count the number of observations above (positive) and below (negative) the hypothesized median, exclude ties (observations equal to the median), and proceed with the same binomial calculation. This is useful for testing whether a population median equals a specific value.