THAT'S MATHS:DOZENS OF RECORDS were smashed at the Beijing Olympics, and the London games should be equally thrilling. Attention is focused on the men's 100 metres. How fast can a sprinter cover 100 metres?
The record is regularly broken, but can this go on or is there an ultimate limit? Mathematical analysis of record times sheds some light on this, although it is impossible to give a definite value for the minimum achievable time.
Since 1977 the International Association of Athletics Federations has required times for the 100 metres to be measured automatically to one hundredth of a second. There have been more than 20 record-breaking times since then, discounting those where drugs were involved or where winds were stronger than two metres per second.
Using a method called extreme value analysis (EVA), the observed records may be used to predict likely developments. EVA is a branch of statistics that enables us to estimate the probability of events beyond the range of any that have previously occurred, using observed data to construct a graph called a distribution function.
One popular distribution function is called the generalised Pareto distribution (GPD). The idea is that the “tail” of this function, the part relating to the extreme values, contains the information we desire.
EVA has been widely used in engineering, finance and earth sciences, to predict floods, insurance losses, rogue waves and, with more dubious reliability, market risks. It has also been used by Reza Noubary of Bloomington University, in Pennsylvania, to predict sporting records.
The GPD contains quantities called parameters, and the observed data are used to estimate the values of these. Then the distribution is used to answer questions such as: “What is the probability of a record time less than 9.5 seconds?”
In August 2009 the Jamaican runner Usain Bolt ran 100 metres at the World Championships in Berlin in 9.58 seconds, breaking his own record by more than a tenth of a second. Using the top six legal record times, ranging from 9.58 to 9.74 seconds, to estimate the parameters of the GPD, we find that the chance of a record time of 9.5 seconds being reached is about 0.5 per cent. But three of the top times were set by Bolt.
Suppose that we omit Bolt’s times. The top six records then range from 9.71 to 9.84 seconds. Using the GPD with parameters estimated from these timings, the probability of a time of 9.5 seconds being broken is then less than 0.25 per cent. So lightning Bolt’s spectacular times have changed our view of what is possible.
But is there an ultimate limit for the 100 metres and, if so, can we find it? We cannot give an exact value and say it will never be beaten; the question can only be answered in terms of probability. We can use the probability distribution to compute a 90 per cent confidence interval for the ultimate record.
Using the top six records, we get a lower bound of 9.4 seconds. Omitting Bolt’s times as before, the computed lower bound is 9.62 seconds, greater than Bolt’s record in Berlin.
So Bolt has already beaten the “ultimate record”, based on the times of other runners. He himself has said that 9.4 seconds is possible.
It seems that Bolt is in another league. Yet, in the Jamaican trials in June, he was beaten into second place by his compatriot Yohan Blake. So who is the “fastest man in the world”? Statistics cannot predict the outcome in London, but at the final, on August 5th, we will have less than 10 seconds to wait to find out.
Peter Lynch is professor of meteorology at University College Dublin
