Nonparametric Tests
Nemenyi Test Table: Critical Values for Comparing Average Ranks After a Friedman Test
Critical values for the Nemenyi post-hoc test, which finds which pairs of groups differ after a significant Friedman test, with a critical difference calculator.
- k = 2 to 20 groups
- α = 0.10 to 0.01
- Critical difference calculator
Nemenyi Test Table
Jump to a table below. Rows: number of groups k · Columns: significance level α · Cells: the first table gives qα = q/√2, used directly in the critical difference; the second gives the studentized range q.
Critical values q/√2 (used in the critical difference)
| Significance level (α) → Groups (k) ↓ | α = 0.1 | α = 0.05 | α = 0.025 | α = 0.01 |
|---|---|---|---|---|
| 2 | 1.645 | 1.960 | 2.241 | 2.576 |
| 3 | 2.052 | 2.344 | 2.604 | 2.913 |
| 4 | 2.291 | 2.569 | 2.817 | 3.113 |
| 5 | 2.460 | 2.728 | 2.968 | 3.255 |
| 6 | 2.589 | 2.850 | 3.084 | 3.364 |
| 7 | 2.693 | 2.948 | 3.177 | 3.452 |
| 8 | 2.780 | 3.031 | 3.256 | 3.526 |
| 9 | 2.855 | 3.102 | 3.324 | 3.590 |
| 10 | 2.920 | 3.164 | 3.383 | 3.646 |
| 11 | 2.978 | 3.219 | 3.435 | 3.696 |
| 12 | 3.030 | 3.268 | 3.482 | 3.741 |
| 13 | 3.077 | 3.313 | 3.525 | 3.781 |
| 14 | 3.120 | 3.354 | 3.564 | 3.818 |
| 15 | 3.159 | 3.391 | 3.600 | 3.853 |
| 16 | 3.196 | 3.426 | 3.634 | 3.884 |
| 17 | 3.230 | 3.458 | 3.665 | 3.914 |
| 18 | 3.261 | 3.489 | 3.694 | 3.941 |
| 19 | 3.291 | 3.517 | 3.721 | 3.967 |
| 20 | 3.319 | 3.544 | 3.747 | 3.992 |
Studentized range q (df = ∞)
| Significance level (α) → Groups (k) ↓ | α = 0.1 | α = 0.05 | α = 0.025 | α = 0.01 |
|---|---|---|---|---|
| 2 | 2.326 | 2.772 | 3.170 | 3.643 |
| 3 | 2.902 | 3.314 | 3.682 | 4.120 |
| 4 | 3.240 | 3.633 | 3.984 | 4.403 |
| 5 | 3.478 | 3.858 | 4.197 | 4.603 |
| 6 | 3.661 | 4.030 | 4.361 | 4.757 |
| 7 | 3.808 | 4.170 | 4.494 | 4.882 |
| 8 | 3.931 | 4.286 | 4.605 | 4.987 |
| 9 | 4.037 | 4.387 | 4.700 | 5.078 |
| 10 | 4.129 | 4.474 | 4.784 | 5.157 |
| 11 | 4.211 | 4.552 | 4.858 | 5.227 |
| 12 | 4.285 | 4.622 | 4.925 | 5.290 |
| 13 | 4.351 | 4.685 | 4.985 | 5.348 |
| 14 | 4.412 | 4.743 | 5.041 | 5.400 |
| 15 | 4.468 | 4.796 | 5.092 | 5.448 |
| 16 | 4.519 | 4.845 | 5.139 | 5.493 |
| 17 | 4.568 | 4.891 | 5.183 | 5.535 |
| 18 | 4.612 | 4.934 | 5.224 | 5.574 |
| 19 | 4.654 | 4.974 | 5.262 | 5.611 |
| 20 | 4.694 | 5.012 | 5.299 | 5.645 |
Tip: click any value to highlight its row and column.
How to read this table
- Run the Friedman test on the k groups over N blocks (subjects, datasets or matched sets). Continue only if it is significant.
- Compute the average rank of each group across the N blocks.
- Read qα from the first table for your k and α, and compute CD = qα √[k(k + 1) / (6N)].
- Two groups differ significantly if their average ranks differ by at least CD.
Worked example
Four methods (A–D) are compared on N = 14 datasets and the Friedman test is significant. Their average ranks are A = 1.7, B = 2.1, C = 2.9, D = 3.3. At α = 0.05 with k = 4, Row k = 4, α = 0.05 gives qα = 2.569.
The critical difference is CD = 2.569 × √[4 × 5 / (6 × 14)] = 1.254. Only the pair A and D (difference 1.6) differ by more than CD, so that pair differs significantly.
Click the underlined links to highlight the value in the table.
Calculator
Enter k, the number of blocks N and α. Optionally paste the average ranks to list the pairs that differ.
Formula and how the values were computed
q is the upper α point of the studentized range distribution with k groups and infinite degrees of freedom, as in Demšar (2006). Dividing by √2 converts it to the form used with the standard error of a difference between average ranks. The k = 2 row equals the normal critical value (for example 1.960 at α = 0.05), as it should.
Frequently asked questions
When should I use the Nemenyi test?
After a Friedman test rejects the hypothesis that all k groups are equivalent over N blocks. The Nemenyi test is the rank-based counterpart of Tukey's HSD for all pairwise comparisons.
What is the difference between the Nemenyi test and Bonferroni-Dunn?
The Nemenyi test compares all pairs and controls the familywise error rate over all k(k − 1)/2 comparisons. The Bonferroni-Dunn test compares each group with a single control and is more powerful for that purpose.
Why infinite degrees of freedom?
The variance of rank differences is known from the ranking itself, not estimated from the data, so the studentized range is used with infinite df.
Is the Nemenyi test conservative?
It is considered conservative, especially when k is large, so real differences can be missed. Some authors prefer Holm- or Shaffer-adjusted pairwise Wilcoxon or Conover tests for more power.