Wilcoxon Signed-Rank Critical Values Table

How to Use the Wilcoxon Signed-Rank Table

Calculate the differences between paired observations and remove zero differences. Rank the absolute values of the remaining differences, then sum the ranks separately for positive and negative differences. For a two-tailed test, use the smaller rank sum as T. Find the number of nonzero pairs in the table and compare T with the appropriate critical value.

Wilcoxon Signed-Rank Test Formula

The signed-rank test uses both the direction and the magnitude ordering of paired differences. After ranking absolute nonzero differences, add the ranks with positive signs and those with negative signs. Their total equals n(n+1)/2 when there are no tied absolute differences and consecutive ranks are used.

dᵢ is the paired difference. W+ = Σ positive ranks; W− = Σ negative ranks; T = min(W+, W−); W+ + W− = n(n+1)/2. Reject the two-sided null when T ≤ Tcritical.

What Sample Size Should You Use?

For this reference table, n is the number of nonzero paired differences, not necessarily the number of original pairs. A pair with an exact difference of zero is omitted under the stated convention before assigning signed ranks.

One-Tailed and Two-Tailed Critical Values

The two-sided test considers departures in either direction and uses the smaller rank sum T under the table’s convention. A one-sided test evaluates a specified direction and must use the corresponding directional rank sum. The chosen tail convention affects the critical value and its interpretation.

Define d = first observation − second observation before looking at the data. For a positive-shift alternative, compare W− ≤ c, equivalently W+ ≥ n(n+1)/2 − c. For a negative-shift alternative, compare W+ ≤ c. Do not substitute min(W+,W−) into a one-sided decision.

Tied Ranks and Zero Differences

The ordinary exact table assumes distinct absolute nonzero differences. If absolute differences are tied, average ranks are commonly assigned, and the null distribution changes. In that situation, a tie-aware exact calculation or another justified method should be used instead of treating the standard critical table as universally exact.

Pairs must be independent of other pairs. The standard signed-rank location interpretation assumes a continuous, symmetric distribution of paired differences. Under the null, signs are independent and equally likely conditional on magnitudes. This table omits zeros (the Wilcox convention), not the Pratt zero-ranking convention. Lookup is exact for n = 5–50, with no normal approximation.

Wilcoxon Signed-Rank vs. Mann–Whitney U

The Wilcoxon signed-rank test uses differences within paired observations. The Mann–Whitney U test compares two independent groups. The names are related, but the designs, rank statistics and critical-value tables are different. Choose the table that matches the study design.