Interpreting results: Normality test

This section explains how normality test can assess whether your data are likely to have been sampled from a Gaussian distribution. Look elsewhere if you want to plot a frequency distribution and for help on deciding when to use nonparametric tests.

Prism offers three normality tests as part of the Column Statistics analysis. These tests require seven or more values, and help you assess whether those values were sampled from a Gaussian distribution.

If you randomly sample from a Gaussian population, what is the probability of obtaining a sample that deviates from a Gaussian distribution as much (or more so) as this sample does?

A small P value is evidence that your data was sampled from a nongaussian distribution. A large P value means that your data are consistent with a Gaussian distribution (but certainly does not prove that the distribution is Gaussian).

Normality tests are less useful than some people guess. With small samples, the normality tests don't have much power to detect nongaussian distributions. With large samples, it doesn't matter so much if data are nongaussian, since the t tests and ANOVA are fairly robust to violations of this standard.

We recommend relying on the D'Agostino-Pearson normality test. It first computes the skewness and kurtosis to quantify how far from Gaussian the distribution is in terms of asymmetry and shape. It then calculates how far each of these values differs from the value expected with a Gaussian distribution, and computes a single P value from the sum of these discrepancies. It is a versatile and powerful normality test, and is recommended. Note that D'Agostino developed several normality tests. The one used by Prism is the "omnibus K2" test.

An alternative is the Shapiro-Wilk normality test. We prefer the D'Agostino-Pearson test for two reasons. One reason is that, while the Shapiro-Wilk test works very well if every value is unique, it does not work well when several values are identical. The other reason is that the basis of the test is hard to understand.

Earlier versions of Prism offered only the Kolmogorov-Smirnov test. We still offer this test (for consistency) but no longer recommend it. It computes a P value from a single value: the largest discrepancy between the cumulative distribution of the data and a cumulative Gaussian distribution. This is not a very sensitive way to assess normality, and we now agree with this statement1:"The Kolmogorov-Smirnov test is only a historical curiosity. It should never be used."1

The Kolmogorov-Smirnov method as originally published assumes that you know the mean and SD of the overall population (perhaps from prior work). When analyzing data, you rarely know the overall population mean and SD. You only know the mean and SD of your sample. To compute the P value, therefore, Prism uses the Dallal and Wilkinson approximation to Lilliefors' method (Am. Statistician, 40:294-296, 1986). Since that method is only accurate with small P values, Prism simply reports “P>0.10” for large P values.

1 RB D'Agostino, "Tests for Normal Distribution" in Goodness-Of-Fit Techniques edited by RB D'Agostino and MA Stepenes, Macel Decker, 1986.