| Interpreting normality tests
How the normality test works
Prism tests for deviations from Gaussian distribution using the Kolmogorov-Smirnov (KS) test. Since the Gaussian distribution is also called the Normal distribution, the test is called a normality test. The KS statistic (which some other programs call D) quantifies the discrepancy between the distribution of your data and an ideal Gaussian distribution - larger values denoting larger discrepancies. It is not informative by itself, but is used to compute a P value.
Prism calculates KS using the method of Kolmogorov and Smirnov. However, the method originally published by those investigators cannot be used to calculate the P value because their method 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.
How to think about results from a normality test
The P value from the normality test answers this question: 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? More precisely, the P value answers this question: If the population is really Gaussian, what is the chance that a randomly selected sample of this size would have a KS value as large as, or larger than, observed?
By looking at the distribution of a small sample of data, it is hard to tell whether or not the values came from a Gaussian distribution. Running a formal test does not make it easier. The tests simply have little power to discriminate between Gaussian and non-Gaussian populations with small sample sizes. How small? If you have fewer than five values, Prism doesn't even attempt to test for normality. But the test doesn't really have much power to detect deviations from Gaussian distribution unless you have several dozen values.
Your interpretation of the results of a normality test depends on the P value calculated by the test and on the sample size.
| P value |
Sample size |
Conclusion |
|
Small
(>0.05)
|
Any |
The data failed the normality test. You can conclude that the population is unlikely to be Gaussian. |
| Large |
Large |
The data passed the normality test. You can conclude that the population is likely to be Gaussian, or nearly so. How large does the sample have to be? There is no firm answer, but one rule-of-thumb is that the normality tests are only useful when your sample size is a few dozen or more.
|
| Large |
Small |
You will be tempted to conclude that the population is Gaussian. But that conclusion may be incorrect. A large P value just means that the data are not inconsistent with a Gaussian population. That doesn't exclude the possibility of a non-Gaussian population. Small sample sizes simply don't provide enough data to discriminate between Gaussian and non-Gaussian distributions. You can't conclude much about the distribution of a population if your sample contains fewer than a dozen values.
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