any assumptions on the distributions of X or Y or the distribution of (X,Y). Under the null hypothesis of independence of X and Y, the sampling distribution of τ has an expected value of zero. The precise distribution cannot be characterized in terms of common distributions, but may be calculated exactly for small samples; for larger samples, it is common to use an approximation to the normal distribution, with mean zero and variance . Accounting for ties A pair is said to be tied if or ; a tied pair is neither concordant nor discordant. When tied pairs arise in the data, the coefficient may be modified in a number of ways to keep it in the range [−1, 1]: Tau-a The Tau-a statistic tests the strength of association of the cross tabulations. Both variables have to be ordinal. Tau-a will not make any adjustment for ties. It is defined as: where nc, nd and n0 are defined as in the next section. Tau-b The Tau-b statistic, unlike Tau-a, makes adjustments for ties. Values of Tau-b range from −1 (100% negative association, or perfect inversion) to +1 (100% positive association, or perfect agreement). A value of zero indicates the absence of association. The Kendall Tau-b coefficient is defined as: where A simple algorithm developed in BASIC computes Tau-b coefficient using an alternative formula. Be aware that some statistical packages, e.g. SPSS, use alternative formulas for computational efficiency, with double the 'usual' number of concordant and discordant pairs. Tau-c Tau-c (also called Stuart-Kendall Tau-c) is more suitable than Tau-b for the analysis of data based on non-square (i.e. rectangular) contingency tables. So use Tau-b if the underlying scale of both variables has the same number of possible values (before ranking) and Tau-c if they differ. For instance, one variable might be scored on a 5-point scale (very good, good, average, bad, very bad), whereas the other might be based on a finer 10-point scale. The Kendall Tau-c coefficient is defined as: where Significance tests When two quantities are statistically independent, the distribution of is not easily characterizable in terms of known distributions. However, for the following statistic, , is approximately distributed as a standard normal when the variables are statistically independent: Thus, to test whether two variables are statistically dependent, one computes , and finds the cumulative probability for a standard normal distribution at . For a 2-tailed test, multiply that