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THE NORMALITY TESTS

= WEEK 9-10 =

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WHY DO WE PERFORM NORMALITY TESTS?

A number of statistical methods/models assume that the samples come from a normally distributed population.
Examples:
One-sample t test for mean
simple and multiple linear regressions

THE METHODS

Visual methods: histogram, the Normal P-P Plot, Box and Whisker plot
Formal tests: Kolmogorov-Smirnov test for normality, Shapiro-Wilk test for normality
The methods are complementary, not one substituting for another.
None of the methods is better than another in an absolute sense.

SOME VISUAL METHODS:�HISTOGRAMS, THE NORMAL P-P PLOTS, BOX AND WHISKER PLOTS

THE FORMAL TESTS:�THE KOLMOGOROV-SMIRNOV AND SHAPIRO-WILK TESTS FOR NORMALITY

H₀: The samples come from a normally distributed population.
If the test is significant (p-value < α), then reject H₀ and conclude that the samples are not from a normally distributed population.
If the test is not significant, the samples fail to indicate that the population where they come from deviates from normality.
It is possible that these formal tests suggest different conclusions. (See the table below.)

GENERAL GUIDELINES/NOTES

There is no type of normality test that can conclude with 100% certainty that the sample comes from a normally distributed population.
“If your sample size is large, don’t use the significance test of normality, don’t even worry too much about normality at all. In small samples, see if your significance test is significant, but don’t be lulled into a false sense of security if the test is not significant.” (Field, 2018)
Formal tests: They have low power if the sample size is small, but are too sensitive to detect deviations from normality if the sample size is large.
“The Shapiro–Wilk test is more appropriate for small sample sizes (<50 samples), although it can also be used for larger sample sizes, while the Kolmogorov–Smirnov test is used for n ≥ 50.” (https://pmc.ncbi.nlm.nih.gov/articles/PMC6350423/)

REFERENCES