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Modern Genetic Analysis Chapter 18. Quantitative Genetics
Some Basic Statistical Notions

To consider the answers to these questions about the most common kinds of genetic variation, we must first examine a number of statistical tools that are essential in the study of quantitative genetics.

Distributions

The outcome of a cross for a Mendelian character can be described in terms of the proportions of the offspring that fall into several distinct phenotypic classes or often simply in terms of the presence or absence of a class. For example, a cross between a red-flowered plant and a white-flowered plant might be expected to yield all red-flowered plants or, if it were a backcross, 1/2 red-flowered plants and 1/2 white-flowered plants. However, we require a different mode of description for quantitative characters. The basic concept is that of the statistical distribution. If the heights of a large number of male undergraduates are measured to the nearest 5 centimeters (cm), they will vary (say, between 145 and 195 cm), but many more individuals will fall into the middle categories (say, between 170 and 180 cm) than at the extremes.

Representing each measurement class as a bar, with its height proportional to the number of individuals in each class, we can graph the result as shown in Figure 18-2a. Such a graph of numbers of individuals observed against measurement class is a frequency histogram. Now suppose that five times as many individuals are measured, each to the nearest centimeter. The classes in Figure 18-2a are now subdivided to produce a histogram like the one shown in Figure 18-2b. If we continue this process, refining the measurement but proportionately increasing the number of individuals measured, then the histogram eventually takes on the continuous appearance of Figure 18-2c, which is the distribution function of heights in the population.

This continuous curve is an idealization, because no measurement can be taken with infinite accuracy or on an unlimited number of individuals. Moreover, the measured variate itself may be intrinsically discontinuous because it is the count of some number of discrete objects such as eye facets or bristles. It is sometimes convenient, however, to develop concepts by using this slightly idealized picture as a shorthand for the more cumbersome observed frequency histogram (Figure 18-2a).

Statistical Measures

Although a distribution contains all of the information about a set of measurements, we would like to be able to distill this information into a few characteristic numbers that convey the necessary information about the distribution without giving it in detail. The characteristics of a distribution that we would like to specify are as follows:

1. Where is the distribution located along the range of possible values? Are the observed values near 100 or near 1000? Therefore we need a measure of central tendency.
2. How much variation is there among individual measurements. Are they all concentrated around the central measurement or do they vary widely across a large range? That is, we need a measure of dispersion.
3. If we are considering more than one characteristic, how are the values of these different characteristics related? Is there any relation between body size and fertility? Is it positive or negative? Do larger parents have larger offspring? If so, we would regard this as evidence that genes influence body size. Thus, we need measures of relation between measurements.

Among the most commonly used measures of central tendency are the most frequent observation, the mode, and the arithmetic average of the observations, the mean. The dispersion of a distribution is almost always measured by the variance, which is average squared distance of the observations from their mean. The relation between different variables is measured by their correlation, which is the average product of the deviation of one variable from its own mean times the deviation of the other variable from its own mean.

These most commonly used measures of central tendency, dispersion, and relation are discussed in detail in the Statistical Appendix to this chapter (page 596). The detailed discussion is put into a separate section in order not to interrupt the flow of logic of the consideration of quantitative genetics. It should not be assumed, however, that an understanding of those statistical concepts is somehow secondary. A proper understanding of quantitative genetics requires a grasp of that material.

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