SUMMARY We present a test of normality based on a statistic D which is up to a constant the ratio of Downton's linear unbiased estimator of the population standard deviation to the sample standard deviation. For the usual levels of significance Monte Carlo simulations indicate that Cornish-Fisher expansions adequately approximate the null distribution of D if the sample size is 50 or more. The test is an omnibus test, being appropriate to detect deviations from normality due either to skewness or kurtosis. Simulation results of powers for various alternatives when the sample size is 50 indicate that the test compares favourably with the Shapiro-Wilk W test, Vbl, b2 and the ratio of range to standard deviation. Shapiro & Wilk (1965) presented a test of normality based on a statistic W consisting essentially of the ratio of the square of the best, or approximately best, linear unbiased estimator of the population standard deviation to the sample variance. They supplied weights for the ordered sample observations needed in computing the numerator of W and also percentile points of the null distribution of W for samples of size 3 to 50. Subsequent investigation (Shapiro, Wilk & Chen, 1968) revealed that this test has surprisingly good power properties. It is an omnibus test, that is, it is appropriate for detecting deviations from normality due either to skewness or kurtosis, which appears to be superior to 'distance' tests, e.g. the chi-squared and Kolmogorov-Smirnov tests. It also usually dominates such standard tests as 1bl, third standardized sample moment; b2, fourth standardized sample moment; and u, ratio of the sample range to the sample standard deviation. Shapiro and Wilk did not extend their test beyond samples of size 50. A number of reasons indicate that it is best not to make such an extension. First, there is the problem of the appropriate weights for the ordered observations for the numerator of W. Each sample size requires a new set. The proliferation of tables is obvious and undesirable. However, even if the appropriate weights were computed from the expected values of the ordered observations from the standardized normal distribution (Harter, 1961), there would still be the uninviting problem of finding the appropriate null distribution of W. Because W's moments beyond the first are unknown, Cornish-Fisher expansions or similar techniques are not applicable. Further, the extension of the normal approximation for W based on Johnson's bounded curves (Shapiro & Wilk, 1968) when the sample is greater than 50 would require an extrapolation and the procedure for implementing it is not available. Simulation runs seem to be the only available way to obtain the null distribution. We present a new test of normality applicable for samples of size 50 or larger which possesses the desirable omnibus property. It requires no tables of weights and for samples of