p. 123 and on pages 372–373 gives the adjusted statistic: and normality is rejected if exceeds 0.631, 0.754, 0.884, 1.047, or 1.159 at 10%, 5%, 2.5%, 1%, and 0.5% significance levels, respectively; the procedure is valid for sample size at least n=8. The formulas for computing the p-values for other values of are given in Table 4.9 on p. 127 in the same book. Tests for other distributions Above, it was assumed that the variable was being tested for normal distribution. Any other family of distributions can be tested but the test for each family is implemented by using a different modification of the basic test statistic and this is referred to critical values specific to that family of distributions. The modifications of the statistic and tables of critical values are given by Stephens (1986) for the exponential, extreme-value, Weibull, gamma, logistic, Cauchy, and von Mises distributions. Tests for the (two-parameter) log-normal distribution can be implemented by transforming the data using a logarithm and using the above test for normality. Details for the required modifications to the test statistic and for the critical values for the normal distribution and the exponential distribution have been published by Pearson & Hartley (1972, Table 54). Details for these distributions, with the addition of the Gumbel distribution, are also given by Shorack & Wellner (1986, p239). Details for the logistic distribution are given by Stephens (1979). A test for the (two parameter) Weibull distribution can be obtained by making use of the fact that the logarithm of a Weibull variate has a Gumbel distribution. Non-parametric k-sample tests Fritz Scholz and Michael A. Stephens (1987) discuss a test, based on the Anderson–Darling measure of agreement between distributions, for whether a number of random samples with possibly different sample sizes may have arisen from the same distribution, where this distribution is unspecified. The R package kSamples and the Python package Scipy implements this rank test for comparing k samples among several other such rank tests. For samples the statistic can be computed as follows under the assumption that the distribution function of -th sample is continuous where is the number of observations in the -th sample is the total number of observations in all samples is the pooled ordered sample is the number of observations in the -th sample that are not greater than . See also Kolmogorov–Smirnov test Kuiper's test Shapiro–Wilk test Jarque–Bera test Goodness