## Batting form and the "hot hand" Russell Degnan

There has been an interesting paper in circulation recently that deals with the idea of a "hot hand", which in cricket terms wed refer to as "good form". It is primarily interesting because for several decades, the idea of streaks being anything other than random luck has been derided. Attempts to measure it in cricket have been few and far between, but there was little to suggest a batsman was more likely to score runs having just done so - or indeed, that they werent largely replicable with a random number generator.

The paper in question upends a key piece of prior research because of a rather simple, but slightly counter-intuitive piece of statistics. The various explanationstend towards the counter-intuitive end of the scale but Ill try to explain.

The technique being used is very elegant: take an action that occurs roughly 50% of the time and measure the number of successes that follow a previous success. If they are completely independent, the subsequent attempt will continue to have a probability of 50%. If the successes are clustered around other successes, that number ought to be higher.

The quirk, is that because strings of attempts are being measured, the average probability found in those strings will not actually be 50%. This is the counter-intuitive part, but is relatively easily seen on a simple graph:

Here is shown strings of eight attempts broken down into the number of times a measure took place (that is, the previous attempt saw a success). The number of instances of measurements breaks down as a binomial distribution centred on 3.5: 2 of zero (ooooooox and oooooooo) and seven (xxxxxxxx and xxxxxxxo), 14 of one and six, and so on as follows:

AttemptsOpportunities usedInstancesPercentage
002
114147.14
2844221.43
32107035.71
42807050.00
52104264.29
6841478.57
714292.86

The distribution of opportunities to take a measurement is similar, but because it takes more successful attempts to generate higher opportunities, it is shifted slightly across, and centred around 4 (or n/2). The breakdown also demonstrates the key to the problem: if four opportunities are to be had, the attempts will be distributed in such a way that the average success rate is 50%. But the only way to generate 7 opportunities is to have succeeded in each of the first 7 measurements. The percentage will be either 6/7 (83.33%) or 7/7 (100%). And as a consequence, the average of multiple strings of measurements ought to sit not at 50% (the middle of the opportunity distribution), but at the centre of the instances of measurement distribution (plus a term for the two extras) - around 45% for strings of length 8.

All very fascinating, particularly as it implies that previous studies showed a "hot hand" after all. But what does it say about cricket? The short answer is that this is a very elegant way of measuring form: find the median score for a batsman, if they surpass it, then test their subsequent score.

For Tendulkar, who played so many innings that the expected percentage is close to 50, his test "form" saw a 53.1% success rate in innings where hed surpassed the median (excluding not outs below the median). In ODIs however (counting only matches where he opened) the figure drops to 50.5%.

That is only a single data point, and some batsmen are likely to be more prone to runs of form than others, but it also points to an issue. In ODI cricket, where multi-lateral series exist, a batsman tends to shuffle opposition quite quickly, and therefore face a reasonable variety of bowling strength from match to match. In test cricket, the subsequent innings is less likely to be independent from the first, without being held in identical conditions - the second innings being on a wearing pitch. Apparent runs of form may just be a string of matches against poor opposition.

Conversely, ODI cricket may be less prone to form, being a format that requires a higher amount of risk-taking, and therefore more luck. Hence a discrepancy between test and ODI matches is feasible. Comparing all innings adds in time gaps when a player might fall out of form (and vice versa), and a proper study ought to remove them. The relative sparsity of innings means that when a player is really in form, it would be hard to distinguish between that and luck with any method. Most likely the effect is small - perhaps three or four runs on a batting average, but probably half that.

Hence measuring the effect, if any, of form remains difficult. On selection matters, - the only avenue where form might matter - there is a lot to be said for judging a player on technique, temperament and overall career trajectory and ignoring runs of form. Everything else is largely academic, albeit an interesting question.

Idle Summers 16th October, 2015 00:45:10   [#]