Assignment 2When the average male of a species differs from the average female, the species is said to be. Humans, for example, are sexually dimorphic with respect to body size,men being on average about 10% taller than women. In this exercise you will analyze dataon sexual dimorphism in non-human primates.Sexual dimorphism can be measured in a variety of ways. In the exercise below, you willuse the following statisticD = log10 (average male weight) log10 (average female weight)where log10 is the logarithm to base 10. For a detailed discussion of this statistic and otherslike it, see the appendix. If you are not interested in a detailed discussion, then just read thefollowing paragraph.A 1-gram difference between males and females would be more significant among micethan among elephants. It makes more sense to give equal weight to equal percentage differ-ences. This is the function of the logarithms in the definition above of D. If males are 1%larger than females, then D = 0:00995 for mice as well as elephants.Answer the questions below using the data in Table 2.1 and Figure 2.1has the largest value of D?(+) and each monogamous species as a circle (). Species without sexual dimorphismshould fall on the 45 line. If males are larger than females, then the species will plotabove this line and the vertical distance above the line measures sexual dimorphism.As an example, the point for the Talapoin Monkey (Miopithecus talapoin) is alreadyplotted for you.P= polygynous; M= monogamous* Data are from Steven J. C. Gaulin and Lee Douglas Sailer. Sexual dimorphism in weight among the primates: The relative impact of allometry and sexual selection. 5(6):515-535, 1984.We all know that10 = 100This same fact can also be expressed by writinglog100 = 2In words, this reads the logarithm to base 10 of 100 equals 2. Both equations mean thesame thing: if you raise 10 to the 2nd power you get 100.Similarly, 10 = 1000, so log 1000 = 3. Here are some other numbers and their logs:Notice the pattern. As grows, log grows too, but much more slowly. . For example, a one-ounce difference is large if we are comparing mice, but small if we are comparing elephants. Thus, we would not want to use ounces as the horizontal axis of a graph that included values for animals ranging in size from mouse to elephant. The graph is easier to interpret if we use log ounces instead, because a proportional difference of (say) 10% between mice occupies the same space on the graph as a 10% difference between elephants. This is why logarithmic scales are used so often in illustrations.How does one measure sexual dimorphism? To make this question concrete, let us taketwo species:Which species is most dimorphic? The answer to this question depends on how we measuresexual dimorphism. One possibility is to simply use the difference between male and female weight values. Using this measure, the gibbon is most dimorphic since males of that species are(on average) 0.32 kg larger than females (5.82-5.5) whereas male talapoin monkeys are only 0.3 kg larger than their females (1.4-1.1).This measure of dimorphism is unsatisfactory, however, because it ignores the overall difference in size between the two species. Surely a 1 gram difference is more important in comparisons among mice than in comparisons among elephants.To incorporate the effect of overall body size, we use a different measure of di-morphism. One possibility is the ratio, R, of male to female size. For the gibbon in ourexample,= = = 1.058female weight 5.5For the talapoin monkey, = 1:27. Thus, male agile gibbons are only 5.8% larger thanfemales, but male talapoin monkeys are 27% larger. By this measure, the monkey is moredimorphic than the gibbon. This seems a more natural measure of sexual dimorphism sinceit automatically scales the result to body size.There is one remaining problem, which arises when we try to make a graph of sexualdimorphism against body size. If the horizontal axis of our graph were measured in (say)kilograms, then two animals that differed by 1 gram would be separated by the same distance on our graph, whether they were mice or elephants. Yet as we argued above, a 1-gramdifference is more significant among small animals than among large ones. To avoid this difficulty, we work with logarithms. (If you are not sure what a logarithmis see Appendix A, above.) In the exercise above, you were asked to make a graph withthe logarithm of average female weight on the horizontal axis and the logarithm of averagemale weight on the vertical axis. If the average male and female weights are equal, thenspecies will fall on a line that is drawn from the origin at an angle of 45. If males are largerthan females, then that species will plot above the 45 line. The vertical distance, D, fromthe 45 line to a speciess point on the graph is equal to = log(average male weight) log(average female weight)For example, in the case of the agile gibbon, the vertical distance to the 45 degree line isD = log 5:82 log 5:5 = 0.765 -0.740 = In the case of the talapoin monkey, the vertical distance to the 45 degree line isD = log 1.4 log 1.1 = 0.146 0.041 = This new quantity, D, is another way to measure sexual dimorphism. Since it is based onlogarithms, it is automatically scaled to body size and will not obscure differences withinsmall species or exaggerate those within large ones.We now have two measures of sexual dimorphism, R and D, both of which take bodysize into account in a natural fashion. It turns out that = log. For example, in the case of the gibbon,D = log R = log 1.058 = We can calculate either as the log of the ratio of male and female sizes or as the differenceof the logs. The answer is the same in either case. The two measures of dimorphism, and, will never disagree. If species A is more dimorphic than species B according to R, then itis also more dimorphic according to D. Thus, both statistics provide the same informationand it does not matter much which we use. In this exercise you have been asked to use rather than because this makes it easier to interpret the graph.