Average Pace

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by Diana Davis


Molly Huddle. Photo by Scott Mason.

Suppose you did a 1.5 mile tempo run in 9:00 (a 6:00 pace). You must have covered some mile of the race in exactly 6:00, right?

Amazingly, the answer is no! My recent math paper, written with some colleagues at Northwestern, explores and answers this question and some related questions about functions. Let’s see why there need not be a mile at the average pace:

For the 1.5 miles in 9:00, if you ran at a constant 6:00 pace, then certainly there would be a mile in exactly 6:00. But if you ran the first and last half miles in 2:00 each, and the middle half mile in 5:00, then you covered every mile in 7:00!

I was inspired to investigate this phenomenon several years ago by Molly Huddle.

In November 2013, Huddle ran 37:49 for 12k, defeating Shalane Flanagan and setting a world best for this distance. However, after the race, LetsRun reported:

Please realize women have run faster for 12k before. Mary Keitany‘s half-marathon world record (21.1 km) of 65:50 would put her going through 12k if evenly paced in 37:26.7.

“If evenly paced” is the key here: Essentially, if she ran fast at the beginning and end, and slow in the middle (the classic beginning runner “race plan”!), then it is possible that every 12k subset was slower than 37:49. See the paper for example with precise times, and a graph.
Here’s another example:

Kenenisa Bekele holds the world records for 5k and 10k, but he never ran a mile race in under 4:00. Or did he? This message board thread explains:

Person 1: Bekele ran a 7:26 3000. That averages under a 4 minute mile for the entire race.

Person 2: Yes, but were there 1760 yards that were consecutively run in under 4 minutes? Otherwise the answer is no, Bekele has never gone sub-4.

Person 3: How would someone run 3000m in 7:26 without ever breaking sub 4-mile pace for a mile portion of the race? Explain that.

Person 4: For an easy example, imagine he ran 1 second for the first 200m, 3:00 for the next 3 laps, 59 seconds for the middle 200m, 3:00 for the next 3 laps, then 1 second for the final 200m. No matter how you slice that, no 1609m is faster than 4:00. That gives a 7:01 3k. Adding time would not create any faster intermediate splits. Realistically, say 25, 60, 60, 60, 36, 60, 60, 60, 25, run evenly for each split.

Here’s how this kind of thing works in general:
  • If your race distance is a whole number of miles, then you must have done some mile in exactly the average pace. (It might be from mile 0.48 to 1.48, for example.)
  • If your race distance is not a whole number of miles, then it’s possible to construct a “race plan” example where no mile is at exactly the average pace.

Don’t ask these guys about pacing. Photo by Scott Mason.

Note that the units “miles” can be changed to anything else — say, 12km in Huddle’s example — and then the rule is based on whether the race distance is a whole number multiple of that distance.

Here’s how the first part works, for those of you who like this sort of thing: Suppose your race is 5 miles and you ran it in 33:00, so an average pace of 6:36. We want to show that you ran some mile in exactly 6:36.
Let’s look at your time for each mile between mile markers, the mile splits recorded on your GPS watch: if they are all less than 6:36, then your total time could not be 33:00 (it would be faster). Similarly, if they are all slower than 6:36, then your total time could not be 33:00 (it would be slower). So either every mile is exactly 6:36, or some mile is slower and some mile is faster than 6:36. If it’s the first case, clearly there is some mile in exactly 6:36, and you’re great at pacing! If it’s the second case, then we can slide over the mile that we’re looking at, and by the Intermediate Value Theorem, somewhere in between, there is a mile at exactly 6:36. For example, if you ran 6:22 for mile 2 and 6:40 for mile 3, then maybe you ran the mile between 2.23 to 3.23 in exactly 6:36.
The second part is harder, because we have to show that we can construct a race plan for any time and any race distance. For more on that, check out the paper.
Wait, there’s more! Here’s another question: Suppose I run the second half of a 10k in a time that’s faster than my 5k PR. There’s no ambiguity here — I looked at my watch at the 5k mark, and at the end of the race, and I covered the last 5k faster than my PR, so I have a new PR. Right?
No again! Purists will point out that every race starts from a standstill. So in order for it to be a true PR, you have to measure it from the gun to the 5k mark. This is why mile races that take a 1500m split for qualifying purposes put the camera at the 1500m mark and take the time from the starting line to there, rather than putting a camera at the 109m mark and taking the time from there to the finish line.
Here’s an amazing application of this phenomenon: The second half of a 200m race is frequently run faster than the runner’s 100m PR, even though the race is twice as far! This is because the slowness from having to accelerate at the beginning makes a lot of difference in such a short race.
Feature image by Kevin Morris.
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