Biostatistics For Dummies, 2nd Edition (Monika Wahi, John Pezzullo)

CHAPTER 9 Summarizing and Graphing Your Data 121

Skewness

Skewness refers to the left-right symmetry of the distribution. Figure 9-3 illus-

trates some examples.

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 consid-

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

How do you express skewness in a summary statistic? The most common skew-

ness 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 4 /

N , 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

FIGURE 9-3:

Distributions

can be left-

skewed (a),

symmetric (b), or

right-skewed (c).