Correlation

Kendall's Tau Table: Exact Critical Values of τ

Exact critical values of Kendall's tau, the rank correlation based on concordant and discordant pairs.

n = 4 to 100
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Kendall's tau Table

Rows: sample size n · Columns: α (one-tailed on the first header row, two-tailed on the second) · Cells: critical τ. Reject H₀ if |τ| ≥ the cell.

Kendall's Tau Table (Critical Values)
One-tailed α →	0.05	0.025	0.01	0.005	0.001
Two-tailed α → n ↓	0.1	0.05	0.02	0.01	0.002
4	1.000	—	—	—	—
5	0.800	1.000	1.000	—	—
6	0.734	0.867	0.867	1.000	—
7	0.620	0.715	0.810	0.905	1.000
8	0.572	0.643	0.715	0.786	0.858
9	0.500	0.556	0.667	0.723	0.834
10	0.467	0.512	0.600	0.645	0.778
11	0.419	0.491	0.564	0.600	0.710
12	0.394	0.455	0.546	0.576	0.667
13	0.359	0.436	0.513	0.565	0.642
14	0.363	0.407	0.473	0.517	0.605
15	0.334	0.391	0.467	0.505	0.581
16	0.317	0.384	0.434	0.484	0.567
17	0.309	0.368	0.427	0.471	0.545
18	0.295	0.347	0.412	0.451	0.530
19	0.287	0.334	0.392	0.439	0.509
20	0.274	0.327	0.379	0.422	0.495
21	0.267	0.315	0.372	0.410	0.486
22	0.265	0.308	0.360	0.394	0.472
23	0.257	0.297	0.352	0.392	0.455
24	0.247	0.290	0.341	0.377	0.450
25	0.240	0.287	0.334	0.367	0.440
26	0.237	0.280	0.330	0.360	0.428
27	0.231	0.271	0.322	0.357	0.419
28	0.228	0.265	0.313	0.344	0.413
29	0.222	0.262	0.311	0.340	0.404
30	0.219	0.256	0.302	0.334	0.394
31	0.213	0.252	0.295	0.325	0.390
32	0.210	0.246	0.291	0.323	0.380
33	0.205	0.243	0.288	0.315	0.375
34	0.202	0.238	0.280	0.312	0.369
35	0.197	0.234	0.278	0.305	0.362
36	0.194	0.232	0.274	0.302	0.359
37	0.193	0.229	0.268	0.298	0.352
38	0.190	0.224	0.264	0.292	0.346
39	0.188	0.220	0.261	0.288	0.342
40	0.185	0.218	0.257	0.285	0.339
41	0.181	0.215	0.254	0.281	0.335
42	0.178	0.213	0.250	0.276	0.329
43	0.177	0.210	0.247	0.274	0.325
44	0.174	0.208	0.244	0.269	0.322
45	0.172	0.205	0.241	0.267	0.318
46	0.170	0.202	0.239	0.264	0.315
47	0.168	0.199	0.236	0.260	0.310
48	0.167	0.197	0.233	0.258	0.307
49	0.164	0.196	0.230	0.254	0.303
50	0.163	0.192	0.228	0.251	0.300
51	0.161	0.191	0.226	0.249	0.298
52	0.159	0.189	0.224	0.246	0.295
53	0.157	0.188	0.221	0.244	0.291
54	0.156	0.186	0.219	0.242	0.288
55	0.155	0.183	0.217	0.240	0.285
56	0.152	0.181	0.215	0.237	0.282
57	0.152	0.180	0.212	0.235	0.280
58	0.150	0.178	0.210	0.232	0.277
59	0.148	0.176	0.209	0.230	0.275
60	0.147	0.175	0.207	0.229	0.273
65	0.141	0.168	0.199	0.219	0.261
70	0.136	0.162	0.191	0.210	0.252
75	0.131	0.156	0.184	0.203	0.243
80	0.126	0.150	0.178	0.197	0.235
85	0.123	0.146	0.172	0.190	0.227
90	0.119	0.141	0.167	0.185	0.221
95	0.115	0.137	0.162	0.180	0.215
100	0.112	0.134	0.158	0.175	0.209

Tip: click any value to highlight its row and column.

How to read this table

Count the concordant (C) and discordant (D) pairs. S = C − D and τ = S / (n(n − 1)/2).
Find the row for n and the column for α.
The association is significant if |τ| is greater than or equal to the table value. To work with S instead, multiply the table value by n(n − 1)/2.

Worked example

Ten patients are ranked by two raters, giving τ = 0.56 (S = 25). Row 10, two-tailed α = 0.05 gives 0.512, so the agreement is significant at the 5% level.

Click the underlined links to highlight the value in the table.

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For values between the rows and columns of the table. Results update as you type.

Formula and how the values were computed

τ = (C − D) / (n(n − 1)/2) · Var(S) = n(n − 1)(2n + 5)/18 under H₀

Values come from the exact null distribution of the number of discordant pairs, which is the distribution of inversions in a random permutation, computed by recursion for every n. Each entry is the smallest attainable τ with P(τ ≥ t) ≤ α.

Frequently asked questions

Kendall's τ or Spearman's rₛ?

Both test the same hypothesis and usually agree. τ is smaller in magnitude (roughly two-thirds of rₛ), has a direct interpretation as a difference between the probabilities of concordance and discordance, and handles ties more gracefully.

What if there are ties?

Use τ-b, which adjusts the denominator for ties. The exact table assumes no ties, so use the normal approximation with a tie-corrected variance when ties are common.

Kendall's Tau Table: Exact Critical Values of τ

Kendall's tau Table

How to read this table

Worked example

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Formula and how the values were computed

Frequently asked questions

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