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The mode Most distributions of phenotypes look roughly like those in Figure 20-3: a single mode is located near the middle of the distribution, with frequencies decreasing on either side. There are exceptions to this pattern, however. Figure 20-18a shows the very asymmetrical distribution of seed weights in the plant Crinum longifolium. Figure 20-18b shows a bimodal (two-mode) distribution of larval survival probabilities for different second-chromosome homozygotes in Drosophila willistoni. A bimodal distribution may indicate that the population being studied could be better considered a mixture of two populations, each with its own mode. 

In Figure 20-18b, the left-hand mode probably represents the subpopulation of severe single-locus mutations that are extremely deleterious when homozygous but whose effects are not felt in the heterozygous state in which they usually exist in natural populations. The right-hand mode is part of the distribution of “normal” viability modifiers of small effect. The mean A more common measure of central tendency is the arithmetic average or the mean. The mean of the measurement ( ) is simply the sum of all the individual measurements (xi) divided by the number of measurements in the sample (N): where represents the operation of summing overall values of i from 1 to N, and xi is the ith measurement. In a typical large sample, the same measured value will appear more than once, because several individuals mean  x  x1  x2  x3 ----- xN N  1 N xi

will have the same value within the accuracy of the measuring instrument. In such a case, can be rewritten as the sum of all measurement values, each weighted by how frequently it occurs in the population. From a total of N individuals measured, suppose that n1 fall in the class with value x1, that n2 fall in the class with value x2, and so forth, so that ni  N. If we let fi be the relative frequency of the ith measurement class, so that , then we can rewrite the mean as where xi equals the value of the ith measurement class. Let us apply these calculation methods to the data of Table 20-3, which gives the numbers of toothlike bristles in the sex combs on the right (x) and left (y) front legs and on both legs (T  x  y) of 20 Drosophila. Looking for the moment only at the sum of the two legs T, we find the mean number of sex comb teeth to be: Alternatively, by using the relative frequencies of the different measurement values, we find that - 13.7 - 0.15(15) - 0.10(16)

Figure 20-19 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 distribution do not describe it completely. They do not distinguish asymmetrical distribution from an asymmetrical one, for example. There are even symmetrical 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 suffices to characterize a distribution. Measures of relationship Covariance and correlation Another statistical notion that is of use in the study of quantitative genetics is the association, or correlation, between variables. 

As a result of complex paths of causation, many variables in nature vary together but in an imperfect or approximate way. Figure 20-20a provides an example, showing the lengths of two particular teeth in several individual specimens of a fossil mammal, Phenacodus primaevis. There is a rough trend such that individuals with longer first molars tend to have longer second molars, but it is considerable scatter of data points around this trend. In contrast, Figure 20-20b shows that the body length and tail length in individual snakes (Lampropeltis polyzoan) are quite closely related to each other, with all the points falling close to a straight line that could be drawn through them from the lower left of the graph to the upper right