Nonparametric Tests
Mann-Kendall Trend Test Table: Critical Values of S for Monotonic Trend
Exact critical values of the Mann-Kendall statistic S for series of 4 to 40 observations, with exact tail probabilities for small n and a calculator with Sen's slope.
- n = 4 to 40
- Exact critical S
- Sen's slope
- Test your data
Mann-Kendall Table
Jump to a table below. First table: rows n · columns one-sided α · cells the smallest S with P(S ≥ s) ≤ α; "–" means no value of S reaches that level. Second table: exact upper-tail probability P(S ≥ s).
Critical values of S (one-sided)
| One-sided α → Series length (n) ↓ | α = 0.1 | α = 0.05 | α = 0.025 | α = 0.01 | α = 0.005 | α = 0.001 |
|---|---|---|---|---|---|---|
| 4 | 6 | 6 | – | – | – | – |
| 5 | 8 | 8 | 10 | 10 | – | – |
| 6 | 9 | 11 | 13 | 13 | 15 | – |
| 7 | 11 | 13 | 15 | 17 | 19 | 21 |
| 8 | 12 | 16 | 18 | 20 | 22 | 24 |
| 9 | 14 | 18 | 20 | 24 | 26 | 30 |
| 10 | 17 | 21 | 23 | 27 | 29 | 35 |
| 11 | 19 | 23 | 27 | 31 | 33 | 39 |
| 12 | 20 | 26 | 30 | 36 | 38 | 44 |
| 13 | 24 | 28 | 34 | 40 | 44 | 50 |
| 14 | 25 | 33 | 37 | 43 | 47 | 55 |
| 15 | 29 | 35 | 41 | 49 | 53 | 61 |
| 16 | 30 | 38 | 46 | 52 | 58 | 68 |
| 17 | 34 | 42 | 50 | 58 | 64 | 74 |
| 18 | 37 | 45 | 53 | 63 | 69 | 81 |
| 19 | 39 | 49 | 57 | 67 | 75 | 87 |
| 20 | 42 | 52 | 62 | 72 | 80 | 94 |
| 21 | 44 | 56 | 66 | 78 | 86 | 102 |
| 22 | 47 | 61 | 71 | 83 | 91 | 109 |
| 23 | 51 | 65 | 75 | 89 | 99 | 115 |
| 24 | 54 | 68 | 80 | 94 | 104 | 124 |
| 25 | 58 | 72 | 86 | 100 | 110 | 132 |
| 26 | 61 | 77 | 91 | 107 | 117 | 139 |
| 27 | 63 | 81 | 95 | 113 | 125 | 147 |
| 28 | 68 | 86 | 100 | 118 | 130 | 156 |
| 29 | 70 | 90 | 106 | 126 | 138 | 164 |
| 30 | 75 | 95 | 111 | 131 | 145 | 171 |
| 31 | 77 | 99 | 117 | 137 | 151 | 181 |
| 32 | 82 | 104 | 122 | 144 | 160 | 188 |
| 33 | 86 | 108 | 128 | 152 | 166 | 198 |
| 34 | 89 | 113 | 133 | 157 | 175 | 207 |
| 35 | 93 | 117 | 139 | 165 | 181 | 215 |
| 36 | 96 | 122 | 146 | 172 | 190 | 226 |
| 37 | 100 | 128 | 152 | 178 | 198 | 234 |
| 38 | 105 | 133 | 157 | 185 | 205 | 243 |
| 39 | 109 | 139 | 163 | 193 | 213 | 253 |
| 40 | 112 | 144 | 170 | 200 | 222 | 264 |
Exact probability P(S ≥ s), n = 4 to 10
| Series length (n) → S ↓ | n = 4 | n = 5 | n = 6 | n = 7 | n = 8 | n = 9 | n = 10 |
|---|---|---|---|---|---|---|---|
| 0 | 0.6250 | 0.5917 | – | – | 0.5476 | 0.5403 | – |
| 1 | – | – | 0.5000 | 0.5000 | – | – | 0.5000 |
| 2 | 0.3750 | 0.4083 | – | – | 0.4524 | 0.4597 | – |
| 3 | – | – | 0.3597 | 0.3863 | – | – | 0.4309 |
| 4 | 0.1667 | 0.2417 | – | – | 0.3598 | 0.3807 | – |
| 5 | – | – | 0.2347 | 0.2810 | – | – | 0.3637 |
| 6 | 0.0417 | 0.1167 | – | – | 0.2742 | 0.3061 | – |
| 7 | – | – | 0.1361 | 0.1907 | – | – | 0.3003 |
| 8 | – | 0.0417 | – | – | 0.1994 | 0.2384 | – |
| 9 | – | – | 0.0681 | 0.1194 | – | – | 0.2422 |
| 10 | – | 0.0083 | – | – | 0.1375 | 0.1792 | – |
| 11 | – | – | 0.0278 | 0.0681 | – | – | 0.1904 |
| 12 | – | – | – | – | 0.0894 | 0.1298 | – |
| 13 | – | – | 0.0083 | 0.0345 | – | – | 0.1456 |
| 14 | – | – | – | – | 0.0543 | 0.0901 | – |
| 15 | – | – | 0.0014 | 0.0151 | – | – | 0.1082 |
| 16 | – | – | – | – | 0.0305 | 0.0597 | – |
| 17 | – | – | – | 0.0054 | – | – | 0.0779 |
| 18 | – | – | – | – | 0.0156 | 0.0376 | – |
| 19 | – | – | – | 0.0014 | – | – | 0.0542 |
| 20 | – | – | – | – | 0.0071 | 0.0223 | – |
| 21 | – | – | – | 0.0002 | – | – | 0.0363 |
| 22 | – | – | – | – | 0.0028 | 0.0124 | – |
| 23 | – | – | – | – | – | – | 0.0233 |
| 24 | – | – | – | – | 0.0009 | 0.0063 | – |
| 25 | – | – | – | – | – | – | 0.0143 |
| 26 | – | – | – | – | 0.0002 | 0.0029 | – |
| 27 | – | – | – | – | – | – | 0.0083 |
| 28 | – | – | – | – | 0.0000 | 0.0012 | – |
| 29 | – | – | – | – | – | – | 0.0046 |
| 30 | – | – | – | – | – | 0.0004 | – |
| 31 | – | – | – | – | – | – | 0.0023 |
| 32 | – | – | – | – | – | 0.0001 | – |
| 33 | – | – | – | – | – | – | 0.0011 |
| 34 | – | – | – | – | – | 0.0000 | – |
| 35 | – | – | – | – | – | – | 0.0005 |
| 36 | – | – | – | – | – | 0.0000 | – |
| 37 | – | – | – | – | – | – | 0.0002 |
| 38 | – | – | – | – | – | – | – |
| 39 | – | – | – | – | – | – | 0.0001 |
| 40 | – | – | – | – | – | – | – |
| 41 | – | – | – | – | – | – | 0.0000 |
| 42 | – | – | – | – | – | – | – |
| 43 | – | – | – | – | – | – | 0.0000 |
| 44 | – | – | – | – | – | – | – |
| 45 | – | – | – | – | – | – | 0.0000 |
Tip: click any value to highlight its row and column.
How to read this table
- Put the observations in time order and compute S = Σ sign(xj − xi) over all pairs i < j.
- For a one-sided test of an upward trend, find the row for n and reject if S is at least the tabulated value (for a downward trend, if S is at most its negative).
- For a two-sided test at level α, use the column for α/2 and compare |S|.
- The values assume no ties. With ties or n above 40, use the normal approximation in the calculator.
Worked example
Twelve annual measurements are 4.1, 4.4, 4.0, 4.9, 5.2, 4.8, 5.6, 5.9, 5.5, 6.3, 6.1, 6.8. Counting, for every later value, whether it is higher (+1) or lower (−1) than each earlier value gives S = 52 out of a maximum of 66.
In the n = 12 row, the one-sided α = 0.025 column gives 30 and the one-sided α = 0.005 column gives 38. Because S = 52 is at least both values, the upward trend is significant at two-sided α = 0.05 and 0.01 (the exact two-sided p-value is 0.0001). Sen's slope is 0.240 units per year.
Click the underlined links to highlight the value in the table.
Calculator
Paste your series in time order. Exact for n up to 40 without ties; otherwise the normal approximation with tie correction.
Formula and how the values were computed
S is a linear function of the number of inversions in a random permutation, S = n(n − 1)/2 − 2 × (inversions), whose exact distribution (the Mahonian numbers) is computed by recursion. Each critical value is the smallest attainable S whose exact upper-tail probability does not exceed α, so tabulated tests are conservative. The exact distribution was checked against full enumeration of all permutations for n = 4 to 8.
Frequently asked questions
What does the Mann-Kendall test test?
It tests for a monotonic upward or downward trend in a series over time, without assuming the trend is linear or that the data are normal. The null hypothesis is that the observations are independent and identically distributed.
How are ties handled?
Tied values add nothing to S, but they reduce its variance. The calculator applies the standard tie correction and uses the normal approximation when ties are present.
Does serial correlation matter?
Yes. Positive autocorrelation makes a trend look significant more often than it should. For autocorrelated series consider pre-whitening, a modified Mann-Kendall test or block bootstrap.
What is the relation to Kendall's tau?
Kendall's tau between the observations and their time index equals S divided by n(n − 1)/2 when there are no ties, so the exact critical values here are the same as for tau against time.