Mean or Median?

Task

Bobbie is a sixth grader who competes in the 100 meter hurdles. In eight track meets during the season, she recorded the following times (to the nearest one hundredth of a second).

$$18.11,\, 31.23,\, 17.99,\, 18.25,\, 17.50,\, 35.55,\, 17.44,\, 17.85$$

1. What is the mean of Bobbie's times for these track meets?  What does this mean tell you in terms of the context?
2. What is the median of Bobbie's times?  What does this median tell you in terms of the context?
3. What information can you gather by comparison of the mean and median?

IM Commentary

The goal of this task is to examine advantages and disadvantages of the mean and median for summarizing a given data set. In this case, the mean is distorted by the two ''outliers'' and so it does not give us an indication of how fast Bobbie is when she runs well and does not get tripped up going over the hurdles. The median gives us a clearer idea of what time Bobbie can expect if she completes the race cleanly. Her coach is likely interested in both: the mean because it gives an indication of how likely Bobbie is to complete the race cleanly while the median gives a clear idea where Bobbie fits, in terms of her speed, relative to other members on the team.

Both the mean and the median are important and, in a situation like this, a single measure of center is not enough. An ideal way to show this data would perhaps be with a box plot which would show both the outliers where Bobbie fell and the rough times she is able to run when she does not fall. The teacher may wish to prompt students to provide a box plot for this data and discuss how it provides more information than just the mean or just the median. 

Solution

1. To find the mean of Bobbie's times, we need to take their sum and divide by the number of races. There are 8 races and their sum is 173.92 seconds, and so we compute the mean by finding 173.92 $\div$ 8 = 21.74. That is, for each second Bobbie completes a race in less than 21.74 seconds, there is a corresponding second Bobbie that completes another race in more than 21.74 seconds.
2. The middle two times for Bobbie's eight races are 17.99 seconds and 18.11 seconds. The average of these two is 36.10 $\div$ 2 = 18.05.  In words, Bobbi completed half the races faster than 18.05 seconds, and half the races slower than 18.05 seconds.
3. That the mean is significantly higher than the median tells us that some of the times slower than 18.05 seconds are much higher than 18.05.  One plausible explanation is that she performs faster than 18.05 seconds at her "standard" pace, but in some races she stumbles and falls on a hurdle, making her finishing time significantly higher.