An Extension of Shapiro and Wilk's W Test for Normality to Large Samples

Abstract

SUMMARY Shapiro and Wilk's (1965) W statistic arguably provides the best omnibus test of normality, but is currently limited to sample sizes between 3 and 50. Wis extended up to n = 2000 and an approximate normalizing transformation suitable for computer implementation is given. A novel application of Win transforming data to normality is suggested, using the three-parameter lognormal as an example. RESEARCH into tests of non-normality was given new impetus with the introduction of the so- called analysis of variance test by Shapiro and Wilk (1965). The test statistic Wwas constructed by considering the regression of ordered sample values on corresponding expected normal order statistics, which for a sample from a normally distributed population is linear; Wwas obtained as an F-ratio from generalized least-squares analysis to judge the adequacy of the linear fit. Percentage points of the null distribution of Wwere tabulated for p = 0 01, 002, 005, 0 1, 0 5, 0-9, 0 95, 0 98 and 0 99, for sample sizes n = 3 (1) 50. A normalizing transformation for Wusing a Johnson SB approximation in the region n = 7(1)50 was later proposed (Shapiro and Wilk, 1968), though tables were still required for n = 4(1) 6. Extensive empirical comparisons of Wwith other tests of non-normality using computer- generated pseudo-random numbers indicated that W had good power properties against a wide range of alternative distributions, and was therefore truly an omnibus test (Shapiro et al., 1968). Subsequently, other statistics of the Wtype, namely Y(D'Agostino, 1971), W' (Shapiro and Francia, 1972) and r (Filliben, 1975), were developed and shown to have power properties broadly comparable with those of W Another approach was to combine ,lb and b2, the