## Quick Reference

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

For *n* > 40, assuming independence, *τ* is approximately an observation from a normal distribution with mean 0 and variance {2(2*n*+5)}/{9*n* (*n*−1)}.

**Critical values for one-tailed tests using τ**

The entries in the table are the smallest values of *τ* (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 *τ*. The asterisk indicates that significance at this level cannot be achieved in this case.

n**5%****1%**n**5%****1%**

41.000*18.294.412

5.8001.00019.287.392

6.733.86720.274.379

7.619.81021.267.371

8.571.71422.264.359

9.500.66723.257.352

10.467.60024.246.341

11.418.56425.240.333

12.394.54526.237.329

13.359.51327.231.322

14.363.47328.228.312

15.333.46729.222.310

16.317.43330.218.301

17.309.42640.185.256

**Critical values for two-tailed tests using τ**

The entries in the table are the smallest values of *τ* (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 *τ*. The asterisks indicate that significance at this level cannot be achieved in these cases.

n**5%****1%**n**5%****1%**

4**18.346.451

51.000*19.333.439

6.8671.00020.326.421

7.714.90521.314.410

8.643.78622.307.394

9.556.72223.296.391

10.511.64424.290.377

11.491.60025.287.367

12.455.57626.280.360

13.436.56427.271.356

14.407.51628.265.344

15.390.50529.261.340

16.383.48330.255.333

17.368.47140.218.285

From *A Dictionary of Statistics* in Oxford Reference.

*Subjects:*
Probability and Statistics.