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Shapes of Distributions

Module by: David Lane

We saw distributions in Distributions that shapes of distributions can differ in skew and/or kurtosis. This part presents numerical indexes of these two measures of shape.

Skew

Figure 1 shows a distribution with a very large positive skew. Recall that distributions with positive skew have tails that extend to the right.

Figure 1: A distribution with a very large positive skew. This histogram shows the salaries of major league baseball players (in thousands of dollars).
Figure 1 (histo2.gif)

Distributions with positive skew have larger means than medians. The mean and median of the baseball salaries shown in Figure 1 are $1,183,417 1,183,417 are and $500,000500,000 respectively. Thus, for this highly-skewed distribution, the mean is more than twice as high as the median. The relationship between skew and the relative size of the mean and median lead the statisticial Pearson to propose the following simple and conventient numerical index of skew: 3Mean-Medianσ 3 Mean Median σ The standard deviation of the baseball salaries is 1,390,922 1,390,922 . Therefore, Pearson's measure of skew for this distribution is 31,183,417-500,0001,390,922=1.47 3 1,183,417 500,000 1,390,922 1.47 .

Just as there are several measures of central tendency, there is more than one measure of skew. Although Pearson's measure is a good one, the following measure is more commonly used. It is sometimes referred to as the third moment about the mean. ∑X-μ3σ3 X μ 3 σ 3

Kurtosis

The following measure of kurtosis is similar to the definition of skew. The value "33" is subtracted to define "no kurtosis" as the kurtosis of a normal distribution. Otherwise, a normal distribution would have a kurtosis of 33. ∑X-μ4σ4-3 X μ 4 σ 4 3

Glossary

Skew:
A distribution is skewed if one tail extends out further than the other. A distribution has positive skew (is skewed to the right) if the tail to the right is longer. See Figure 1 for an example. A distribution has a negative skew (is skewed to the left) if the tail to the left is longer. See Figure 2 for an example.
Figure 2
Figure 2 (skewfig2.png)
Kurtosis:
Kurtosis measures how fat or thin the tails of a distribution are relative to a normal distribution. It is comonly definied as: ∑X-μ4σ4-3 X μ 4 σ 4 3

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