© John Wiley & Sons, Inc.

FIGURE 9-3: Distributions can be left-skewed (a), symmetric (b), or right-skewed (c).

Figure 9-3b shows a symmetrical distribution. If you look back to Figures 9-2a and 9-2c, which are

also symmetrical, they look like the vertical line in the center is a mirror reflecting perfect symmetry,

so these have no skewness. But Figure 9-2b has a long tail on the right, so it is considered right

skewed (and if you flipped the shape horizontally, it would have a long tail on the left, and be

considered left-skewed, as in Figure 9-3a).

How do you express skewness in a summary statistic? The most common skewness coefficient, often

represented by the Greek letter γ (lowercase gamma), is calculated by averaging the cubes (third

powers) of the deviations of each point from the mean and scaling by the SD. Its value can be positive,

negative, or zero.

Here is how to interpret the skewness coefficient (γ):

A negative γ indicates left-skewed data (Figure 9-3a).

A zero γ indicates unskewed data (Figures 9-2a and 9-2c, and Figure 9-3b).

A positive γ indicates right-skewed data (Figures 9-2b and 9-3c).

Notice that in Figure 9-3a, which is left-skewed, the γ = –0.7, and for Figure 9-3c, which is right-

skewed, the γ = 0.7. And for Figure 9-3b — the symmetrical distribution — the γ = 0, but this almost

never happens in real life. So how large does γ have to be before you suspect real skewness in your

data? A rule of thumb for large samples is that if γ is greater than

, your data are probably

skewed.

Kurtosis

Kurtosis is a less-used summary statistic of numerical data, but you still need to understand it. Take a

look at the three distributions shown in Figure 9-4, which 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.