Kurtosis

Kurtosis is a measure of the thickness of the tails of a statistical distribution and the sharpness of its peak. Kurtosis is also called the fourth moment about the mean and is one of the two most common statistics used to describe the shape of a distribution (the other is skewness).

There are three types of kurtosis: mesokurtosis, platykurtosis, and leptokurtosis. Many times, these are referred to as zero, negative, and positive kurtosis, respectively. The positive and negative descriptors refer to whether the peak of the distribution is ‘sharper’ or higher than a normal distribution or if the peak is ‘flatter’ or lower than the normal distribution.

Kurtosis statistics compare the distribution under study with the normal distribution. A distribution that [Page 581]resembles the normal distribution in terms of relative peakedness of the distribution is said to be mesokurtic or have zero kurtosis. If a distribution has a higher peak where the width of the peak is thinner and the tails are thinner than the normal distribution, then the distribution is said to be leptokurtic or have positive kurtosis. If a distribution has a lower peak where the width of the peak is wider and the tails are thicker than the normal distribution, then the distribution is said to be platykurtic or have negative kurtosis. A platykurtic distribution may even have a concave peak instead of a rounded peak (see Figure 1).

Figure 1 Common Distributions With Type of Kurtosis

When analyzing data using an analysis package, a researcher needs to know whether the program uses the kurtosis statistic or the kurtosis excess statistic. For the kurtosis statistic, a value of 3 indicates a normal distribution. However, kurtosis excess is a measure of how far the kurtosis statistic is from 3. So a normal distribution has a kurtosis excess of 0. Using the kurtosis excess statistic, the sign of the statistic matches the description of the kurtosis (−1 is negative kurtosis, + 1 is positive kurtosis).

The most commonly known distributions and their type of kurtosis are given in Table 1.

Many times kurtosis is used to help assess whether a distribution being studied meets the normality assumptions of most common parametric statistical tests. While the normal distribution has a kurtosis of +3, it is important to realize that, in practice, the kurtosis statistic for a sample from the population will not be exactly equal to + 3. How far off can the statistic be and not violate the normality assumption? Provided the statistic is not grossly different from +3, then that decision is up to the researcher and his or her opinion of an acceptable difference. For most typically sized (small) samples, the kurtosis statistic is unreliable. To accurately measure the kurtosis of a distribution, sample sizes of several hundred may be needed.