# Sign Test Calculator

Free sign test calculator. Enter paired differences to perform a non-parametric test of whether the population median equals a hypothesized value.

## What this calculates

Perform the sign test, a non-parametric hypothesis test that evaluates whether the median of paired differences is significantly different from a hypothesized value. No normality assumption is required.

## Inputs

- **Number of Differences** — min 2, max 10 — Number of paired differences (2-10).
- **Hypothesized Median** — The median value under the null hypothesis.
- **Significance Level** — options: 0.01, 0.05, 0.10 — Type I error rate.
- **Difference 1** — Paired difference value.
- **Difference 2** — Paired difference value.
- **Difference 3** — Paired difference value.
- **Difference 4** — Paired difference value.
- **Difference 5** — Paired difference value.
- **Difference 6** — Paired difference value.
- **Difference 7** — Paired difference value.
- **Difference 8** — Paired difference value.
- **Difference 9** — Paired difference value.
- **Difference 10** — Paired difference value.

## Outputs

- **Positive Signs** — Count of differences above the hypothesized median.
- **Negative Signs** — Count of differences below the hypothesized median.
- **Ties (zeros)** — Differences equal to the hypothesized median (excluded).
- **Test Statistic (S)** — The smaller of positive and negative sign counts.
- **P-Value (approx)** — Two-tailed p-value from binomial distribution.
- **Result** — formatted as text — Whether to reject the null hypothesis.

## Details

The sign test is one of the simplest non-parametric tests for matched-pair data. For each observation, you compute the difference from a hypothesized median and record whether it is positive or negative. Ties (differences of zero) are excluded. Under the null hypothesis that the true median equals the hypothesized value, the number of positive signs follows a Binomial(n, 0.5) distribution.

The test statistic S is the smaller of the positive and negative sign counts. A very small S indicates that the data is overwhelmingly on one side of the hypothesized median, providing evidence against the null hypothesis. The two-tailed p-value is computed as twice the cumulative binomial probability of observing S or fewer successes.

The sign test trades statistical power for robustness. It makes no assumptions about the shape of the distribution and works well even with ordinal data or small samples. However, it is less powerful than the Wilcoxon signed-rank test (which also uses the magnitudes of differences) and much less powerful than the paired t-test when the normality assumption holds.

## Frequently Asked Questions

**Q: When should I use the sign test instead of a t-test?**

A: Use the sign test when your data is ordinal rather than interval, when the sample size is very small, or when you cannot assume the differences are normally distributed. It is a safe choice when parametric assumptions are questionable.

**Q: How are ties handled in the sign test?**

A: Observations where the difference equals the hypothesized median exactly (ties) are removed from the analysis. The test is then based only on the remaining non-zero differences. If many ties exist, the effective sample size may be too small for a meaningful result.

**Q: What is the difference between the sign test and the Wilcoxon signed-rank test?**

A: The sign test uses only the direction (positive or negative) of differences, while the Wilcoxon signed-rank test also considers the magnitude of the differences. The Wilcoxon test is more powerful when differences come from a symmetric distribution, but the sign test is valid under weaker assumptions.

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Source: https://vastcalc.com/calculators/statistics/sign-test
Category: Statistics
Last updated: 2026-04-21