Skewness

Implications of a Skewed Distribution

The importance of skewness to parametric statistics involves the assumption of a normal distribution for variables when conducting tests (e.g., correlations, comparing means). The departure from symmetry provides a distortion from the assumptions of the test and may change the level of Type I (false positive) or Type II (false negative) error. While most statistics remain extremely robust to such violations, a formal statistical examination may be warranted.

Alternatively, the existence of a skewed distribution may require the consideration of whether or not some other curve form should be considered as a better representation of the available scores. Most statistical packages will provide for an examination and evaluation of the nature of the distribution.

Testing for Skewness in a Normal Distribution

There are many different tests that can be conducted to test for the issues of skewness and whether the distribution departs from an expected normal distribution (e.g., Anderson-Darling, Cramer, Kolmogorov-Smirnov, Ryan-Joiner, Shapiro-Wilk, Lilliefors). Most computer programs (e.g., SPSS, SAS) will test for the level of skewness as a default function or will provide it as an option.

The tests typically assume that the set of estimates or scores are normally distributed. The test measures the departure of scores from what would be expected in a normal distribution. The null hypothesis assumes a normal distribution. If the test is significant beyond that which is expected due to random chance (typically an Alpha or Type I error rate of 5%, where p < .05) then the distribution sample is considered non-normal or skewed.

Researchers are capable of mathematically assessing the nature and amount of skewness in a data set. A typical test of skewness is found in the following equation:

$$1/n\left[\mathrm{\Sigma }{(\mathrm{X}-\mathrm{\mu })}^3\right]/{\left\{1/n\left[\mathrm{\Sigma }{(\mathrm{X}-\mathrm{\mu })}^2\right]\right\}}^{3/2}$$

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