Solution to Exercise 1.22.

  Problem Statement: How does the speed of a runner vary over the course of a marathon (a distance of 42.195 km.)? Consider determining both the time to run the first 5 km. and the time to ran between the 35-km and 40-km points, and then subtracting the former time from the latter time. A positive value of this difference corresponds to a runner slowing down toward end of the race. The accompanying histogram based on times of runners who participated in several different Japanese marathons.

What are some interesting features of this histograrn? What is a typical difference value? Roughly what proportion of the runners ran the late distance more quickly than the early distance?

Solution: The histogram is clearly skewed right - pretty strongly. There may be a few outliers around 700 seconds, although that is arguable. For a typical value: the mode is in the 50-100 range; the mean is somewhat larger (because it is skewed right), maybe around 200-250 (just a guess). The median would be in between. There seem to be only about 10 runners with negative values for the distance. It's hard to tell, but the sample size n is clearly above 500, and maybe 1,000 on the outside. Thus, the proportion of runners ran the late distance more quickly than the early distance is around 1 to 2 % (10/1000 to 10/500).