Normality should be checked, because many statistical methods used and discussed later in subsequent tutorials assume normality, meaning that the data sample comes from a Gaussian data population. Normality should be checked both visually and numerically. Visually, normality can be assessed with a
q–q plot that plots observed values (observed quantiles) against their quantiles that are implied by a normal distribution. Instead of plotting observed quantiles against implied normal quantiles, one can also plot them directly against their implied standardized normal scores; see
Figure 3. If the data are normally distributed, points on a
q–q plot will exhibit linearity. Furthermore, the slope of the plotted line reflects the standard deviation, and the value where the line intersects with the vertical line at zero provides the mean. In summary, for normal distributions the normal
q–q plots should be linear. Deviations from the linear pattern provide evidence that the underlying distribution is not normal. A
q–q plot is effective because the human eye is quite good at recognizing linear tendencies. For further discussion, see Chapter 2 of Box et al.
12 Widely used programs such as R and Prism provide
q–
q plots along with the various tests for deciding how well a data distribution follows a normal, Gaussian distribution.
Numerically, normality can tested through one of the numerous significance tests for normality, such as the Anderson–Darling normality test, Shapiro–Francia normality test, Lilliefors (Kolmogorov–Smirnov) normality test, Cramer–von Mises normality test, Pearson χ2 normality test, Shapiro–Wilk test for normality, Jarque–Bera normality test, and D’Agostino normality test. Some tests require a minimum number of data points. A probability value is given for how likely the distribution is normally distributed; for example, a probability value of less than 0.05 would mean that there is a significant chance that the distribution is not normal. It should be mentioned that a probability value of 0.05 is commonly used as a criterion level for statistical significance, but this is arbitrary and is, in reality, an oversimplification. Examination of the data distribution using the q–q plot gives one a much better idea of how well the data distribution follows a normal distribution.
Unfortunately, for small samples the visual checks are typically not very informative, and the normal probability tests are not very powerful. Furthermore, because tests quantify deviations from normality using different methods, it is not surprising that they lead to somewhat different results. Not every test is equally sensitive to one or the other violations of normality. Although there is only one normality, there are certainly many different ways of violating normality. For an evaluation of normal probability tests, see Yap and Sim.
13 Prism prefers the D'Agostino omnibus test among the three tests (Kolmogorov–Smirnov, Shapiro–Wilk, and D'Agostino) that it considers.
14
Figure 3 illustrates normal
q–
q plots for the data from our illustrative example. Normality must be checked separately for each of the three groups, as groups have different means and variances. Minor deviations from linearity can be noticed in the plot for the EAE group, with points in the lower and upper tail suggesting a distribution with “heavier” tails than the normal. This can be visualized in the dot plot for the EAE group shown in
Figure 1, where there are more points at the upper and lower portions of the distribution than expected for a Gaussian distribution. The probability values of the normal probability tests in the
Table show that the deviations from normality are only borderline significant.
One needs to keep in mind that no natural distribution is actually normal. As George Box
12 pointed out: “All models are wrong, but some are useful.” If the sample size is big enough, one will always fail a normality test. This is the reason why we encourage researchers to actually look at plots rather than just relying on a probability value. Graphs can tell whether the deviation from normality is substantial enough to cause worry and whether transformations can make a distribution closer to normal.