If one goal of a triathlon is to provide a balanced test of prowess over its three disciplines many of them don’t succeed. Last weekend, for example, Paula and her sister Adrienne participated in the Taunton Deane Sprint triathlon in Wellington, while Zoe did the Novice event. In both events the results were dominated by performance in the cycling alone – places in the Novice were 95% correlated with how competitors placed in the bike segment, and in the Sprint the correlation was 96%. From a competitive point of view, the events were essentially wet bike races.

There’s no mystery about why this is the case. Look at this chart of each competitor’s times for each of the three legs of the Sprint:

The fastest swimmer gained under 10 minutes from the slowest (see the blue dots), while the fastest cyclist made over 41 minutes over the most recreational bike performance (red dots). The run times (green dots) are intermediate between the two, with the spread between fastest and slowest being around 26 minutes. This is why a good cyclist will always beat an equivalently good swimmer (or runner) on a course like this, other things being equal.

This would be easy to fix by making the swim leg (and, if necessary, the run) longer relative to the bike ride. Moreover, we can work out * exactly* how much to change the length of each leg so that (1) the variation in times for most competitors in each of the three disciplines is the same, and (2) a competitor in the middle of the pack comes home in the same time after the leg lengths have been re-balanced. In fact, triathlons are lucky in this regard as three is the only number of events for which there is one and only one way to rebalance the legs to achieve these two goals precisely.

To work out how to do this, we need to decide on a model for how changing the length of each event will change the finishing times of the competitors. You could wrestle over the right way to do this for long time; for example, increasing the length of the swim might not only slow down the pace of swimming but could also make the running slower too. But however you model it there will be one solution for the new leg lengths. For simplicity, I assumed that adjusting the lengths of the legs doesn’t materially affect the pace at which the participants tend to perform.

Doing the maths (see “Tech corner” below), to make the event fair while keeping the mid-pack finishing time the same, we find that the distances would have to change as follows:

- The
**swim**increases from 400m to just over 1,050m - The
**bike**ride is reduced from 23k to just under 16k - The
**run**stays about the same at 5k.

Calculating the impact this would have on each competitor shows that Adi would move up six places and Paula would move up seven. Running the same analysis on the Novice event, shows that Zoe would move up two places. However, it would be a disaster for our friend Brendon who is a good cyclist but a reluctant swimmer: he would fall from 5th to 25th place.

If we re-draw the chart above to show how performances in the Sprint would be adjusted if the leg lengths were re-balanced we get this:

The bike segment still, on average, takes longer than the run, which itself averages out longer than the swim. However, the * distribution* of times is now equal between all three making it a fair contest between the disciplines. The same procedure can be applied to any triathlon and modified to fit any set of assumptions about the impact of adjusting leg lengths on performance. Why don’t they do it?

**Tech corner**

My condition that the variation of times for the three events be the same is achieved by requiring that they each have the same standard deviation. This is better than pegging the spread from fastest to slowest to be the same because it neutralises the effect of exceptional (fast or slow) individuals or events (e.g. multiple punctures).

The condition that competitors in the middle of the pack should come home in the same time after the lengths are adjusted as they did in the actual event is achieved by requiring the sum of median times for each discipline to be the same after as before.

I copied all of the ride data from the results website. (If you use Excel to do this beware of how its formatting mangles times as they straddle the hour point.)

My assumption that changing the leg distances doesn’t materially affect the pace makes it easy to get an explicit formula for the new lengths (see below). Different models will generate different equations. I also assume that you can set each leg length to anything you like but in real life this will be constrained. For example, if you have to swim whole lengths in a pool and run/bike complete circuits round a set course you end up with Diophantine equations; but having a name with four syllables doesn’t stop the equations being amenable to attack, especially using a tool like Excel that has a built-in solver.

Personally, I use R rather than Excel whenever I can.

With my constant pace model you calculate the new lengths as follows…

Set the median swim, bike and run times to mS, mB and mR respectively. (Use MEDIAN() in Excel.)

Set the standard deviation of the swim, bike and run times to sS, sB and sR respectively. (Use STDEV() in Excel.)

Then calculate multipliers for the three legs:

kS = (mS+mB+mR)/(mS+mB.sS/sB+mR.sS/sR)

kB = (mS+mB+mR)/(mS.sB/sS+mB+mR.sB/sR)

kR = (mS+mB+mR)/(mS.sR/sS+mB.sR/sB+mR)

Then multiply the swim, bike and run lengths respectively by these three factors respectively. Likewise, the individual rider times in each discipline are multiplied by the corresponding factor. Transition times, of course, are unchanged.

[…] I’ve shown before that triathlons disadvantage relatively strong swimmers and favour relatively strong cyclists. This is also evident from the distribution of results at Wimbleball as the above stats indicate and these charts more conclusively show: […]

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Wimbleball 70.3 2013 Stats | Hill Farm Houseon June 17, 2013at 9:27 pm

[…] was hoping to do some analysis of the ride data, as I’ve done for previous triathlons before. However, the results information on the event website is abysmal. There is no file download, no […]

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A few photos from the Virgin Active London Triathlon « Hill Farm Houseon September 24, 2012at 7:08 pm

[…] how the organisers could re-balance the legs in future years to make the event fairer, you can use the formula that I gave before. Share this:PrintEmailLike this:LikeBe the first to like […]

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Dartford Bridge Tri results digest « Hill Farm Houseon June 25, 2012at 4:34 pm

Triathletes are the smartest athletes so it seems. Interesting analysis.

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Dr. James Johnsonon May 19, 2012at 9:35 pm