Compare signs above, below, or across pairs. See ties, exact probabilities, confidence levels, and charts. Make defensible nonparametric decisions using transparent calculations and exports.
This paired example compares two measurements for the same ten cases.
| Case | Sample A | Sample B | Difference | Sign |
|---|---|---|---|---|
| 1 | 12 | 10 | 2 | + |
| 2 | 15 | 14 | 1 | + |
| 3 | 18 | 19 | -1 | - |
| 4 | 9 | 8 | 1 | + |
| 5 | 14 | 12 | 2 | + |
| 6 | 16 | 15 | 1 | + |
| 7 | 20 | 18 | 2 | + |
| 8 | 11 | 11 | 0 | 0 |
| 9 | 13 | 14 | -1 | - |
| 10 | 17 | 15 | 2 | + |
The sign test converts each difference into a direction.
For paired data, compute d = A - B.
For one-sample testing, compute d = x - m0.
If d > 0, assign a positive sign. If d < 0, assign a negative sign. If d = 0, treat it as a tie and remove it.
Let n be the number of non-tied observations. Let X be the number of positive signs.
Under the null hypothesis, X follows a binomial distribution:
X ~ Binomial(n, 0.5)
Two-sided exact p-value:
p = 2 × P(Binomial(n, 0.5) ≤ min(X, n - X))
Upper-tail exact p-value:
p = P(Binomial(n, 0.5) ≥ X)
Lower-tail exact p-value:
p = P(Binomial(n, 0.5) ≤ X)
The large-sample approximation uses:
z = (X - n/2 ± 0.5) / √(n/4)
The interval shown on this page is a Wilson interval for the positive sign proportion.
It tests whether positive and negative differences are balanced. In paired work, it checks whether the median difference is zero. In one-sample work, it checks whether the population median matches a chosen reference value.
Use it when data are paired, ordinal, skewed, or resistant to strong distribution assumptions. It is useful when only direction matters more than the size of the difference.
Ties are observations with zero difference. The sign test removes them because they do not support either direction. The usable sample size becomes the count of non-tied observations.
The exact value comes from the binomial distribution and is preferred for small and moderate samples. The normal approximation is a convenient large-sample estimate and serves as a backup summary.
It is the share of non-tied observations with positive differences. Under the null hypothesis, that proportion should be near 0.5. Strong imbalance suggests evidence against the null.
Yes. Enter the before values in Sample A and the after values in Sample B, or reverse the order if your alternative direction needs it. Interpretation follows the sign of A minus B.
No. It only uses direction. A very large positive difference and a tiny positive difference both count as one positive sign. That makes the method robust but less informative than rank-based tests.
It means the observed sign imbalance is not strong enough at the chosen alpha level. It does not prove the null hypothesis is true. It only means the sample evidence is insufficient.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.