Sunday, 10 June 2012

Maths of the Heptathlon: Why the scoring system is flawed

The Heptathlon is one of the greatest tests of an all-round athlete that exists in World Athletics. Women compete in seven events over two days for the title. But this sport is suffering because of a biased scoring system. Why is this, and what can be done?

Let's start with a bit about the Heptathlon. It has seven events, three running events (100 m Hurdles, 200m and 800m), two jumping events (High Jump and Long Jump), and two throwing events (Shot Put and Javelin). Certain equations are then used to turn the athletes' raw scores into points, which are then totalled, and whoever has the most points wins.

The current scoring system was developed by Dr Karl Ulbrich in 1952, and amended in 1984. He based these early scoring systems on certain benchmarks: certain results were adjudged to be worth 1000 points, and others worth 0 points. A line was then put through these, but it was an upwards curve rather than a straight line, to account for the fact that the better an athlete performs, the harder it is to better that score by a certain amount. In other words, it is much easier for an athlete to reduce a time for the 100m Hurdles from 13.5s to 13.0s than from 13.0s to 12.5s.

He used three different formulae to calculate the score, P for each event.

P = a(b-T)c
for the running events, where T is the time in seconds.
P = a(M-b)c
for the jumping events, where M is the heigh/length in cm.
P = a(D-b)c
for the throwing events, where D is the heigh/length in metres.

a, b and c are different for each event, and are given in the following table.

200 meters4.9908742.51.81
800 meters0.111932541.88
100 metres hurdles9.2307626.71.835
High Jump1.8452375.01.348
Long Jump0.1888072101.41
Shot Put56.02111.501.05
Javelin Throw15.98033.801.04
This system works in principle, but a look at modern results begins to reveal a problem. Here is a table of the highest and lowest scores in each individual event at the Heptathlon at the 2011 World Championships in Daegu.

Times given in seconds (minutes:seconds for 800m), and distances in metres. The difference is in points.
A first glance at this suggests that it is biased towards the the Hurdles, the scores for this event seem to be much higher than others! However, how high or low the scores are is actually irrelevant. For example, if you decided to add 100 points to everyone's javelin score, the outcome of the competition would still be exactly the same, but with everyone's scores simply 100 points higher. 

The important statistic is instead how large the spread of scores is, here shown by the difference between the highest and lowest points scored. For example, the best hurdler at the Championships only gained 201 points over the worst hurdler, but the best javelin thrower gained a huge 400 points over the worst! In this sense, with the current scoring, the Javelin is worth almost twice as much as the Hurdles, and the Shot Put is worth nearly as much, leaving specialist throwers with a great advantage.

These differences are not just due to individual weak points giving very poor, anomalous scores. In the table below are the 10th place results for each event, and the distance from first.

At the top end, the competition is even more biased towards the Javelin, which is nearly 3 times as important as the Hurdles when considering only the top 10 results.

In order to have a truly fair system, and to provide the best test of who is really the top all-rounder, the difference between any two given positions should be as close as possible for each event. 

The values for b have been carefully chosen to set the result that will score 0 points, and c has been chosen to give the graph the right shape. These values seem to work, and should not be changed. However, we can change 'a' to try and make the difference between the best and worst athletes equal. Using the data here, we can find new values for 'a' for each sport.

Let us set the difference between the best and worst results for each discipline to be 300 points. Taking the Hurdles as an example, we can form the following equation.

a(b-Tbest)c = a(b-Tworst)c + 300

a(b-Tbest)c - a(b-Tworst)c = 300
then substitute in the values.
a(26.7-12.93)1.835 - a(26.7-14.32)1.835 = 300
a(13.77)1.835 - a(12.38)1.835 = 300
Then factorise the left side of the equation to get
a(13.771.835 - 12.381.835) = 300
21.8198368a = 300

Using the same method, we can find new values of a for all other disciplines to give a difference of 300. These values turn out to be the following:

Using this scoring system, our table from before turns out like this.

The only problem with this is that the scores have radically different values awarded to them. This doesn't impact on fairness (except in the rare event of three fouls or false starts in any event), but could be very confusing to athletes and spectators, and may also affect competitors psychologically. To fix this, we can introduce a new value, d, which is added to the end of each formula, to change the highest score from each of these disciplines to 1050 and the lowest to 750.

This also improves the differences between the top ten.

The Standard Deviation of the difference between 1st and 10th place is 13.4 with the system we propose, compared to 61.3 with the old system. This is clearly a much better and fairer method, and one we hope the IAAF will consider. To sum up, here is the Proposed Scoring System in full.

For running events:
P = a(b-T)c + d
For jumping events:
P = a(M-b)c + d
For throwing events:
P = a(D-b)c + d

And the following table of values are used:

Unfortunately, with the current scoring system, British Gold hopeful Jessica Ennis is at an acute disadvantage. Her best events, Hurdles and High Jump, are two of the most under-awarded events, and her weak point, Javelin, is the most over-awarded! Jessica came 2nd at these Championships in Daegu by a margin of 127 points, with only her Javelin below par on this occasion, but with these scoring revisions would have lost by a much tighter margin of 42 points. With the current scoring system, it seems that Jessica, along with all the other athletes, should turn their focus towards the Shot Put and Javelin in hope of securing Gold.

We're all behind you Jess!

We hope you enjoyed this post, it was certainly very interesting to research and write. We don't have any Olympic Heptathlon tickets, but will be watching Jess with earnest.

Theo Caplan

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Check out our last two posts:
What is the Drake Equation? (Guest post from B.C.) - A guest post from one of our partners, about the equation used to predict our chances of discovering alien civilisations. Can maths answer the age-old question of whether or not we are alone?
The Physics of Gymnastics - The forces exerted on a gymnasts body during some routines are extreme, so how does it work?

1 comment:

  1. Excellent work...keep it up such an interesting insight and really hope they take your proposal for the new scoring system...they need to listen to our youngest and brightest...regards James - Engineer by trade, Physicist by hobby