Nonparametric Tests
Friedman Test Table: Exact Critical Values
Exact critical values for the Friedman test, the nonparametric alternative to repeated-measures ANOVA.
- k = 3 to 6 treatments
- Exact values
- Exact calculator
Friedman Test Table
Jump to a value of k below. Rows: number of blocks (subjects) · Columns: α · Cells: critical Friedman statistic. Reject H₀ if the statistic ≥ the cell.
k = 3 treatments
| α → n (blocks) ↓ | 0.1 | 0.05 | 0.025 | 0.01 | 0.005 | 0.001 |
|---|---|---|---|---|---|---|
| 3 | 6.000 | 6.000 | — | — | — | — |
| 4 | 6.000 | 6.500 | 8.000 | 8.000 | 8.000 | — |
| 5 | 5.200 | 6.400 | 7.600 | 8.400 | 10.000 | 10.000 |
| 6 | 5.334 | 7.000 | 8.334 | 9.000 | 10.334 | 12.000 |
| 7 | 5.429 | 7.143 | 7.715 | 8.858 | 10.286 | 12.286 |
| 8 | 5.250 | 6.250 | 7.750 | 9.000 | 9.750 | 12.250 |
| 9 | 5.556 | 6.223 | 8.000 | 9.556 | 10.667 | 12.667 |
| 10 | 5.000 | 6.200 | 7.800 | 9.600 | 10.400 | 12.600 |
| 11 | 5.091 | 6.546 | 7.819 | 9.455 | 10.364 | 13.273 |
| 12 | 5.167 | 6.500 | 8.000 | 9.500 | 10.500 | 12.667 |
| 13 | 4.770 | 6.616 | 7.539 | 9.385 | 9.847 | 12.462 |
| 14 | 5.143 | 6.143 | 7.429 | 9.143 | 10.429 | 13.286 |
| 15 | 4.934 | 6.400 | 7.600 | 8.934 | 10.534 | 12.934 |
| 16 | 4.875 | 6.500 | 7.625 | 9.375 | 10.125 | 13.500 |
| 17 | 5.059 | 6.118 | 7.412 | 9.295 | 10.706 | 13.059 |
| 18 | 4.778 | 6.334 | 7.445 | 9.000 | 10.334 | 13.000 |
| 19 | 5.053 | 6.422 | 7.685 | 9.579 | 10.211 | 13.369 |
| 20 | 4.900 | 6.300 | 7.500 | 9.300 | 10.800 | 13.300 |
k = 4 treatments
| α → n (blocks) ↓ | 0.1 | 0.05 | 0.025 | 0.01 | 0.005 | 0.001 |
|---|---|---|---|---|---|---|
| 2 | 6.000 | 6.000 | — | — | — | — |
| 3 | 6.600 | 7.400 | 8.200 | 9.000 | 9.000 | — |
| 4 | 6.300 | 7.800 | 8.400 | 9.600 | 10.200 | 11.100 |
| 5 | 6.360 | 7.800 | 8.760 | 9.960 | 10.920 | 12.600 |
| 6 | 6.400 | 7.600 | 8.800 | 10.200 | 11.400 | 12.800 |
| 7 | 6.429 | 7.800 | 9.000 | 10.543 | 11.400 | 13.458 |
| 8 | 6.300 | 7.650 | 9.000 | 10.500 | 11.550 | 13.800 |
| 9 | 6.200 | 7.667 | 8.867 | 10.734 | 11.800 | 14.067 |
| 10 | 6.360 | 7.680 | 9.000 | 10.680 | 11.880 | 14.520 |
| 11 | 6.273 | 7.691 | 9.000 | 10.746 | 12.055 | 14.564 |
| 12 | 6.300 | 7.700 | 9.100 | 10.800 | 12.100 | 14.800 |
| 13 | 6.139 | 7.800 | 9.093 | 10.847 | 12.231 | 14.908 |
| 14 | 6.343 | 7.715 | 9.086 | 10.886 | 12.258 | 15.086 |
| 15 | 6.280 | 7.720 | 9.160 | 10.920 | 12.200 | 15.080 |
k = 5 treatments
| α → n (blocks) ↓ | 0.1 | 0.05 | 0.025 | 0.01 | 0.005 | 0.001 |
|---|---|---|---|---|---|---|
| 2 | 7.200 | 7.600 | 8.000 | 8.000 | — | — |
| 3 | 7.467 | 8.534 | 9.600 | 10.134 | 10.667 | 11.467 |
| 4 | 7.600 | 8.800 | 9.800 | 11.200 | 12.000 | 13.200 |
| 5 | 7.680 | 8.960 | 10.240 | 11.680 | 12.480 | 14.400 |
| 6 | 7.734 | 9.067 | 10.400 | 11.867 | 13.067 | 15.200 |
| 7 | 7.772 | 9.143 | 10.515 | 12.115 | 13.258 | 15.658 |
| 8 | 7.700 | 9.200 | 10.600 | 12.300 | 13.500 | 16.000 |
| 9 | 7.734 | 9.245 | 10.667 | 12.445 | 13.689 | 16.356 |
| 10 | 7.760 | 9.280 | 10.720 | 12.480 | 13.840 | 16.560 |
k = 6 treatments
| α → n (blocks) ↓ | 0.1 | 0.05 | 0.025 | 0.01 | 0.005 | 0.001 |
|---|---|---|---|---|---|---|
| 2 | 8.286 | 9.143 | 9.429 | 9.715 | 10.000 | — |
| 3 | 8.715 | 9.858 | 10.810 | 11.762 | 12.524 | 13.286 |
| 4 | 9.000 | 10.286 | 11.429 | 12.715 | 13.572 | 15.286 |
| 5 | 9.000 | 10.486 | 11.743 | 13.229 | 14.258 | 16.429 |
| 6 | 9.048 | 10.572 | 12.000 | 13.620 | 14.762 | 17.048 |
Tip: click any value to highlight its row and column.
How to read this table
- Rank the k treatments within each block (subject) and add up the ranks Rⱼ for each treatment over the n blocks.
- Compute Q = 12/(nk(k + 1)) Σ Rⱼ² − 3n(k + 1).
- Find the table for k and the row for n.
- Reject H₀ if Q is greater than or equal to the table value. Beyond the table, compare Q with chi-square on k − 1 df.
Worked example
Eight patients each try three formulations and rank them, giving Q = 6.75. In the k = 3 table, row 8, α = 0.05 gives 6.250, so the formulations differ at the 5% level.
Click the underlined links to highlight the value in the table.
Calculator
For values between the rows and columns of the table. Results update as you type.
Formula and how the values were computed
Values come from the exact permutation distribution of Q, built block by block over all k! rank orderings. Each entry is the smallest attainable Q with P(Q ≥ q) ≤ α, rounded up. The table assumes no ties within blocks.
Frequently asked questions
What is a block?
A block is a set of related observations, one per treatment: usually one subject measured under every condition, as in a crossover trial.
Friedman or Page's L?
Friedman's test detects any difference among treatments. When you expect a specific order, such as increasing dose, Page's L test is more powerful.