122 PART 3 Getting Down and Dirty with Data

all have the same mean and the same SD. Also, all three have perfect left-right

symmetry, meaning they are unskewed. But their shapes are still very different.

Kurtosis is a way of quantifying these differences in shape.

A good way to compare the kurtosis of the distributions in Figure 9-4 is through

the Pearson kurtosis index. The Pearson kurtosis index is often represented by the

Greek letter k (lowercase kappa), and is calculated by averaging the fourth powers

of the deviations of each point from the mean and scaling by the SD. Its value can

range from 1 to infinity and is equal to 3.0 for a normal distribution. The excess

kurtosis is the amount by which k exceeds (or falls short of) 3.

One way to think of kurtosis is to see the distribution as a body silhouette. If you

think of a typical distribution function curve as having a head (which is near the

center), shoulders on either side of the head, and tails out at the ends, the term

kurtosis refers to whether the distribution curve tends to have»

» ^{A pointy head, fat tails, and no shoulders, which is called leptokurtic, and is}

shown in Figure 9-4a (where k

3).»

» ^{An appearance of being normally distributed, as shown in Figure 9-4b}

(where k

3).»

» ^{Broad shoulders, small tails, and not much of a head, which is called}

platykurtic. This is shown in Figure 9-4c (where k

3).

A very rough rule of thumb for large samples is that if k differs from 3 by more than

8 /

N , your data have abnormal kurtosis.

FIGURE 9-4:

Three

distributions:

leptokurtic (a),

normal (b), and

platykurtic (c).

© John Wiley & Sons, Inc.