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# ρ Critical Values for Spearman's Rank Correlation Coefficient

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It is assumed that at least one ranking consists of a random permutation of the numbers 1 to n.

For n > 40, assuming independence, ρ is approximately an observation from a normal distribution with mean 0 and variance 1/(n − 1).

Critical values for one-tailed tests using ρ The entries in the table are the smallest values of ρ (to three decimal places) that correspond to one-tail probabilities≤5% (or 1%). The observed value is significant if it is equal to, or greater than, the value in the table. The exact significance level never exceeds the nominal value (5% or 1%). The table can also be used to provide 10% and 2% critical values for two-tailed tests for ρ. The asterisk indicates that significance at this level cannot be achieved in this case.

n5%1%n5%1%

41.000*18.401.550

5.9001.00019.391.535

6.829.94320.380.522

7.714.89321.370.509

8.643.83322.361.497

9.600.78323.353.486

10.564.74524.344.476

11.536.70925.337.466

12.503.67826.331.457

13.484.64827.324.449

14.464.62628.318.441

15.446.60429.312.433

16.429.58230.306.425

17.414.56640.264.368

Critical values for two-tailed tests using ρ

The entries in the table are the smallest values of ρ (to three decimal places) that correspond to two-tail probabilities≤5% (or 1%). The observed value is significant if it is equal to, or greater, than the value in the table. The exact significance level never exceeds the nominal value (5% or 1%). The table can also be used to provide 2.5% and 0.5% critical values for one-tailed tests for ρ. The asterisks indicate that significance at this level cannot be achieved in these cases.

n5%1%n5%1%

4**18.472.600

51.000*19.460.584

6.8861.00020.447.570

7.786.92921.436.556

8.738.88122.425.544

9.700.88323.416.532

10.648.79424.407.521

11.618.75525.398.511

12.587.72726.390.501

13.560.70327.383.492

14.538.67928.375.483

15.521.65429.368.475

16.503.63530.362.467

17.488.61840.313.405

From A Dictionary of Statistics in Oxford Reference.

Subjects: Probability and Statistics.

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