Statistical Tables/Spearman's Rank Correlation Critical Values

Regression & Correlation

Spearman's Rank Correlation Critical Values

The smallest Spearman's rho that is statistically significant for your number of pairs, with exact values for small samples.

Critical values of ρ (Spearman), by number of pairs n

n	Two-tailed α (one-tailed α in grey)
n	0.1 (0.05)	0.05 (0.025)	0.02 (0.01)	0.01 (0.005)
4	1.000	—	—	—
5	0.900	1.000	1.000	—
6	0.829	0.886	0.943	1.000
7	0.714	0.786	0.893	0.929
8	0.643	0.738	0.833	0.881
9	0.600	0.700	0.783	0.833
10	0.564	0.648	0.745	0.794
11	0.536	0.618	0.709	0.755
12	0.503	0.587	0.678	0.727
13	0.484	0.560	0.648	0.703
14	0.464	0.538	0.626	0.679
15	0.446	0.521	0.604	0.654
16	0.429	0.503	0.582	0.635
17 *	0.413	0.486	0.567	0.619
18 *	0.400	0.471	0.551	0.602
19 *	0.389	0.458	0.536	0.586
20 *	0.378	0.446	0.522	0.571
21 *	0.369	0.435	0.509	0.558
22 *	0.360	0.425	0.498	0.545
23 *	0.352	0.415	0.487	0.533
24 *	0.344	0.406	0.476	0.522
25 *	0.336	0.398	0.467	0.512
26 *	0.330	0.390	0.457	0.502
27 *	0.323	0.382	0.449	0.493
28 *	0.317	0.375	0.441	0.484
29 *	0.311	0.368	0.433	0.476
30 *	0.306	0.362	0.426	0.468
35 *	0.283	0.335	0.394	0.433
40 *	0.264	0.313	0.368	0.405
45 *	0.248	0.294	0.347	0.382
50 *	0.235	0.279	0.329	0.363
60 *	0.214	0.254	0.301	0.332
70 *	0.198	0.235	0.278	0.307
80 *	0.185	0.220	0.260	0.287
90 *	0.174	0.207	0.245	0.271
100 *	0.165	0.197	0.233	0.257

Rows n = 4 to 16 are exact: each is found by listing the value of ρ for every possible ordering of the ranks (up to 16! ≈ 2.1 × 10¹³ orderings, counted with a dynamic program rather than one by one) and finding the smallest ρ whose upper-tail probability is at most α ÷ 2. Rows marked * (n ≥ 17) are approximate, from an Edgeworth series using ρ's exact null variance and fourth moment; at n = 16, where both methods can be compared, it lands within 0.003 of the exact value in every column. A dash means no attainable value is significant at that α. Your result is significant if |ρ| is greater than or equal to the tabled value.

How to use this tool

Count the number of pairs, n, you ranked.
Pick the column for your significance level — two-tailed unless you predicted the direction of the relationship in advance.
Read the critical value where your n row meets that column.
If the absolute value of your rho is at least the tabled value, the rank correlation is statistically significant.

Formula Reference

Spearman's rho (no tied ranks)

ρ = 1 − 6Σd² / [n(n² − 1)]

Null variance of rho

Var(ρ) = 1 / (n − 1)

Exact critical value

the smallest ρ with P(ρ ≥ value | no association) ≤ α ÷ 2

ArticleSpearman's Rank Correlation ArticleNon-Parametric Tests: An Introduction CalcCorrelation Calculator TablePearson Correlation Critical Values GlossaryGlossary: Spearman's Rank Correlation

Frequently Asked Questions

Why can't I just use the Pearson correlation table for Spearman's rho?+

For small samples the two null distributions differ noticeably. With 10 pairs, Spearman's rho needs 0.648 to be significant at α = 0.05 (two-tailed), while the Pearson table gives 0.632 for the same sample size. The gap shrinks as n grows.

Why are some cells a dash?+

With very few pairs there are so few possible rankings that even perfect agreement is not unusual enough. With 4 pairs, only 1 of the 24 possible orderings gives ρ = 1, a probability of about 0.042 — below 0.05 one-tailed, but not below 0.025, so nothing is significant at α = 0.05 two-tailed.

Which rows are exact?+

Rows for n = 4 to 16 were computed by counting every possible ordering of the ranks. Beyond that the counting becomes impractical, so rows marked with an asterisk use an Edgeworth approximation, which matches the exact values to within 0.003 at n = 16 and becomes more accurate as n increases.

What if my data have tied ranks?+

Ties are given the average of the ranks they span, which changes the null distribution slightly from the untied one tabulated here. For a few ties the table is still a good guide; with many ties, compute rho as the Pearson correlation of the ranks and use software for the p-value.

n	Two-tailed α (one-tailed α in grey)
4	1.000	—	—	—
5	0.900	1.000	1.000	—
6	0.829	0.886	0.943	1.000
7	0.714	0.786	0.893	0.929
8	0.643	0.738	0.833	0.881
9	0.600	0.700	0.783	0.833
10	0.564	0.648	0.745	0.794
11	0.536	0.618	0.709	0.755
12	0.503	0.587	0.678	0.727
13	0.484	0.560	0.648	0.703
14	0.464	0.538	0.626	0.679
15	0.446	0.521	0.604	0.654
16	0.429	0.503	0.582	0.635
17 *	0.413	0.486	0.567	0.619
18 *	0.400	0.471	0.551	0.602
19 *	0.389	0.458	0.536	0.586
20 *	0.378	0.446	0.522	0.571
21 *	0.369	0.435	0.509	0.558
22 *	0.360	0.425	0.498	0.545
23 *	0.352	0.415	0.487	0.533
24 *	0.344	0.406	0.476	0.522
25 *	0.336	0.398	0.467	0.512
26 *	0.330	0.390	0.457	0.502
27 *	0.323	0.382	0.449	0.493
28 *	0.317	0.375	0.441	0.484
29 *	0.311	0.368	0.433	0.476
30 *	0.306	0.362	0.426	0.468
35 *	0.283	0.335	0.394	0.433
40 *	0.264	0.313	0.368	0.405
45 *	0.248	0.294	0.347	0.382
50 *	0.235	0.279	0.329	0.363
60 *	0.214	0.254	0.301	0.332
70 *	0.198	0.235	0.278	0.307
80 *	0.185	0.220	0.260	0.287
90 *	0.174	0.207	0.245	0.271
100 *	0.165	0.197	0.233	0.257

Two-tailed α (one-tailed α in grey)

0.1 (0.05)

0.05 (0.025)

0.02 (0.01)

0.01 (0.005)

1.000

—

0.900

1.000

—

0.829

0.886

0.943

1.000

0.714

0.786

0.893

0.929

0.643

0.738

0.833

0.881

0.600

0.700

0.783

0.833

0.564

0.648

0.745

0.794

0.536

0.618

0.709

0.755

0.503

0.587

0.678

0.727

0.484

0.560

0.648

0.703

0.464

0.538

0.626

0.679

0.446

0.521

0.604

0.654

0.429

0.503

0.582

0.635

17 *

0.413

0.486

0.567

0.619

18 *

0.400

0.471

0.551

0.602

19 *

0.389

0.458

0.536

0.586

20 *

0.378

0.446

0.522

0.571

21 *

0.369

0.435

0.509

0.558

22 *

0.360

0.425

0.498

0.545

23 *

0.352

0.415

0.487

0.533

24 *

0.344

0.406

0.476

0.522

25 *

0.336

0.398

0.467

0.512

26 *

0.330

0.390

0.457

0.502

27 *

0.323

0.382

0.449

0.493

28 *

0.317

0.375

0.441

0.484

29 *

0.311

0.368

0.433

0.476

30 *

0.306

0.362

0.426

0.468

35 *

0.283

0.335

0.394

0.433

40 *

0.264

0.313

0.368

0.405

45 *

0.248

0.294

0.347

0.382

50 *

0.235

0.279

0.329

0.363

60 *

0.214

0.254

0.301

0.332

70 *

0.198

0.235

0.278

0.307

80 *

0.185

0.220

0.260

0.287

90 *

0.174

0.207

0.245

0.271

100 *

0.165

0.197

0.233

0.257

How to use this tool

Count the number of pairs, n, you ranked.

Pick the column for your significance level — two-tailed unless you predicted the direction of the relationship in advance.

Read the critical value where your n row meets that column.

If the absolute value of your rho is at least the tabled value, the rank correlation is statistically significant.

Frequently Asked Questions

Why can't I just use the Pearson correlation table for Spearman's rho?+

Why are some cells a dash?+

Which rows are exact?+

What if my data have tied ranks?+