Chapter 4: Analysing the Data Part II : Descriptive Statistics

# Determining if skewness and kurtosis are significantly non-normal

Skewness. The question arises in statistical analysis of deciding how skewed a distribution can be before it is considered a problem. One way of determining if the degree of skewness is "significantly skewed" is to compare the numerical value for "Skewness" with twice the "Standard Error of Skewness" and include the range from minus twice the Std. Error of Skewness to plus twice the Std. Error of Skewness. If the value for Skewness falls within this range, the skewness is considered not seriously violated.

For example, from the above, twice the Std. Error of Skewness is 2 X .183 = .366. We now look at the range from Š0.366 to + .366 and check whether the value for Skewness falls within this range. If it does we can consider the distribution to be approximately normal. If it doesnÕt (as here), we conclude that the distribution is significantly non-normal and in this case is significantly positvely skewed.

Kurtosis. Another descriptive statistic that can be derived to describe a distribution is called kurtosis. It refers to the relative concentration of scores in the center, the upper and lower ends (tails), and the shoulders of a distribution (see Howell, p. 29). In general, kurtosis is not very important for an understanding of statistics, and we will not be using it again. However it is worth knowing the main terms here.

A distribution is platykurtic if it is flatter than the corresponding normal curve and leptokurtic if it is more peaked than the normal curve.

The same numerical process can be used to check if the kurtosis is significantly non normal. A normal distribution will have Kurtosis value of zero. So again we construct a range of "normality" by multiplying the Std. Error of Kurtosis by 2 and going from minus that value to plus that value. Here 2 X .363 = .726 and we consider the range from Š0.726 to + 0.726 and check if the value for Kurtosis falls within this range. Here it doesnÕt (12.778), so this distribution is also significantly non normal in terms of Kurtosis (leptokurtic).

Note, that these numerical ways of determining if a distribution is significantly non-normal are very sensitive to the numbers of scores you have. With small sets of scores (say less than 50), measures of skewness and kurtosis can vary widely from negative to positive skews to perfectly normal and the parent population from which the scores have come from could still be quite normal. Numerical methods should be used as a general guide only.

Modality. A distribution is called unimodal if there is only one major "peak" in the distribution of scores when represented as a histogram. A distribution is "bimodal" if there are two major peaks. If there are more than two major peaks, weÕd call the distribution multimodal. An example of a bimodal distribution can be found in Figure 4.6.

Figure 4.6 An example of a bimodal distribution. The figure shows the frequency of nicotine use in the data base used for Assignment II. Nicotine use is characterised by a large number of people not smoking at all and another large number of people who smoke every day.