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Modern Genetic Analysis Chapter 18. Quantitative Genetics
Measures of Dispersion: The Variance

A second characteristic of a distribution is the width of its spread around the central class. Two distributions with the same mean might differ very much in how closely the measurements are concentrated around the mean. The most common measure of variation around the center is the variance, which is defined as the average squared deviation of the observations from the mean, or

When more than one individual has the same measured value, the variance can be written

To avoid subtracting every value of x separately from the mean, we can use an alternative computing formula that is algebraically identical with the preceding equation:

Because the variance is in squared units (square centimeters, for example), it is common to take the square root of the variance, which then has the same units as the measurement itself. This square-root measure of variation is called the standard deviation of the distribution:

The data in Table 18-2 can be used to exemplify these calculations:

We can use the computing formula that avoids taking individual deviations:

and

Figure 18-14 shows two distributions having the same mean but different standard deviations (curves A and B) and two distributions having the same standard deviation but different means (curves A and C).

The mean and the variance of a distribution do not describe it completely. They do not distinguish a symmetric distribution from an asymmetric one, for example. We can even construct symmetric distributions that have the same mean and variance but still have somewhat different shapes. Nevertheless, for the purposes of dealing with most quantitative genetic problems, the mean and variance suffice to characterize a distribution.

© 1999 by W. H. Freeman and Company. All rights reserved.