$22.$ How does the speed of a runner vary over the course of

a marathon $($a distance of $42\mathrm{.}195$ km$)?$ Consider determining

both the time to run the first $5$ km and the time to

run 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 the end of the race. The accompanying

histogram is based on times of runners who participated

in several different Japanese marathons $($Factors

Affecting Runners$$ Marathon Performance,$$ Chance,

Fall, $1993$: $2430).$

What are some interesting features of this histogram?

What is a typical difference value? Roughly what proportion

of the runners ran the late distance more quickly

than the early distance?

$0100200400$

$50$

$100$

$150$

$200$

$100$

Time

difference