Wilcoxon Signed-Rank Critical Values (T)

The Wilcoxon signed-rank test is the nonparametric alternative to the paired t-test. This table gives its critical values of T: compute T, the smaller of the two signed-rank sums, and find the row for n, the number of pairs with a non-zero difference. If T is less than or equal to the table value, the result is significant at that level. A dash (—) means no result can be significant at that level with so few pairs.

Exact critical values: the largest T whose two-tailed probability under the null hypothesis is at most α, computed by enumerating the exact null distribution of the signed-rank statistic (no normal approximation). Values agree with standard published tables. Column headings are two-tailed α; the one-tailed level is shown underneath.

How to calculate T by hand

Ten people rate their sleep quality (0–100) before and after a two-week routine change:

1. Compute each difference (after − before) and drop any pairs with a difference of exactly zero. Here none are zero, so n = 10.
2. Rank the absolute differences from smallest (1) to largest. Tied absolute values share the average of their ranks.
3. Add up the ranks separately for positive and negative differences: T⁺ = 52 and T⁻ = 1 + 2 = 3. As a check, T⁺ + T⁻ = n(n + 1)/2 = 55.
4. Take the smaller sum, T = 3, and compare it with the table: for n = 10, the critical value is 8 at α = 0.05 and 3 at α = 0.01. Since 3 ≤ 3, sleep quality improved significantly even at the 1% level (exact two-tailed p = 0.0098).

Report the median difference with the test result, for example: “Sleep scores increased after the change (median difference +5.5 points), Wilcoxon signed-rank T = 3, n = 10, p = 0.010.”

When to use the Wilcoxon signed-rank test

• You have paired measurements: the same people before and after, or matched pairs.
• The differences are clearly non-normal or contain outliers, or the data are ordinal, such as ratings, so a paired t-test is doubtful.
• The differences are roughly symmetric around their median. If they are very skewed, the simpler sign test is safer.

To run the test on your own data with an exact p-value, use the Wilcoxon signed-rank test calculator. For two independent groups, use the Mann–Whitney U test instead.

Larger samples

Beyond about 50 pairs, T is close to normal with mean n(n + 1)/4 and standard deviation √(n(n + 1)(2n + 1)/24), so you can compute z = (T − mean) / SD and compare it with ±1.96 for a two-tailed test at 5%. With tied ranks, the standard deviation needs a small correction, which statistical software applies automatically.

Frequently asked questions

How do you use the Wilcoxon signed-rank table?

Compute T, the smaller of the sums of ranks for positive and negative differences, and find the row for n, the number of non-zero differences. If T is less than or equal to the table value, the result is significant.

What is the critical value for n = 10 at 0.05?

For a two-tailed test at 0.05 with 10 pairs, the critical value of T is 8.

Why do I drop zero differences?

A zero difference carries no information about direction, so the standard test removes those pairs and reduces n accordingly.

Is the Wilcoxon signed-rank test the same as the Mann-Whitney test?

No. The signed-rank test is for paired data, while the Mann-Whitney U test (also called the Wilcoxon rank-sum test) is for two independent groups.

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