Tendulkar was great – but the stats say he was not unique

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Sachin Tendulkar, who retired from cricket last week, was the finest batsman of our age. But the greatest batsman of all time remains Don Bradman. The Australian was not just better than his contemporaries. He was so much better that, during his career, top-class cricket was dominated by the issue of how to contain him. The notorious outcome was the “bodyline” Test match series of 1933, when the English bowlers aimed at opposing players rather than the stumps.

Can there be any objective basis for such judgments of relative skill? The most plausible approach is to fit a statistical distribution to the range of sporting achievement, as recorded by some measure such as a batting average. This can be used to predict the relative frequencies of different performance levels. The most appropriate tool for a phenomenon such as sporting achievement is probably a power law.

Power laws stand as one of the two main approaches to quantitative description of the incidence of natural and economic phenomena. These laws seem to be relevant when outcomes can be represented as the random aggregation of chance events (such as the distribution of batting averages). They are useful descriptions when there are many small and a few large outcomes (as with cricket innings). For example, a power law explains the distribution of word frequencies in a text and the incidence of natural disasters. (Typhoon Haiyan, like Mr Tendulkar, is exceptional but likely to be repeated.)

On this basis, cricketers as gifted as Mr Tendulkar will appear from time to time: but it is unlikely that one as skilful as Bradman will appear in the entire history of the world. Those who saw Bradman play just happen to have been lucky that, in their lifetime, one such player did. We shall not see his like again.

We could even use this approach to ask a question such as: “Was Bradman a better cricketer, relative to other contemporary cricketers, than Bobby Fischer was a chess player, relative to other contemporary chess players?” Or even to make comparisons over time. Emil Zatopek, the Czech runner who took all the long-distance golds in the 1952 Olympics, was recently named the greatest track athlete ever by Runner’s World magazine. But was he a more remarkable runner, for his time, than the UK’s Mo Farah is in ours? (Farah is about 90 seconds, or 5 per cent, faster over 10,000 metres than Zatopek). Perhaps the question: “Was Mozart a finer composer than Shakespeare a playwright?” is not as ridiculous or unanswerable as it seems.

The power law seems to have been discovered by Vilfredo Pareto, the Italian economist, at the start of the 20th century. It is the basis of the 80/20 principle, the claim that 20 per cent of the population typically accounts for 80 per cent of income (and many other things).

Normal, or Gaussian, distributions, are more relevant for phenomena that represent the repetition of similar chance events. Such outcomes are clustered around a central tendency. These distributions account for different estimates of the same thing, such as the value of a Twitter share or the speed of a car. They tell us the frequencies in games of chance. If power laws account for the spread of human incomes and talents, normal distributions explain the range of our heights and weights.

In the early 20th century, the French mathematician, Louis Bachelier, used the assumptions of normality and classical statistics to describe the behaviour of the prices of speculative assets, such as stock values. That approach became – and still is – central to mainstream financial economics.

More recently, another French mathematician, Benoit Mandelbrot, attacked the same questions using a power law and the methods of complex systems. Although standard finance theory resists Mandelbrot’s approach, it is followed by many quantitative analysts who have come to finance from maths and physics.

Who is right? The answer is significant because the choice affects substantially the incidence of extreme events, and may also have large consequences for the expected long-term rate of return. Tendulkars and Bradmans, Mozarts and Shakespeares, have an influence on our lives entirely disproportionate to their numbers.

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