Published 2000 Revised 2010

This is a problem about world cricket ratings. Like all sports rankings, cricket ratings involve some maths. In this case, they use a mathematical technique known as exponential weighting. For those who want to know more, read on.

Suppose there are two cricketers, whose scores in five matches have been as follows:

Match 1 | Match 2 | Match 3 | Match 4 | Match 5 | |
---|---|---|---|---|---|

Atherton | 10 | 20 | 30 | 40 | 50 |

Hussain | 50 | 40 | 30 | 20 | 10 |

(The most recent match was match 5.)

Both of these players have scored 150 runs in five matches, so their average score per match is 150/5 = 30.

However, this statistic doesn't say anything about the two players' form. If you look at the trend in Atherton's scores, they have been going steadily upwards, while Hussain's have been going down. Their averages might be the same, but Atherton has the better recent form.

To measure recent form, you could simply take, say, the last two matches and work out the averages for those. (Atherton would have an average of 90/2 = 45, and Hussain would have an average of 30/2 =15). However, there are several ways to produce an average that takes account of all the performances, but gives more credit to the more recent ones.

The cricket ratings use what is called an exponential average. This is how it works.

Suppose you want to treat each match as you go back in time as 10% less important. We will call 10% the 'decay factor'. To work out the exponential average, do the following:

Atherton | Score (S) | Decay (k) | Weighted value, W(S x k) |
---|---|---|---|

Match 5 | 50 | 1.00 | 50.00 |

Match 4 | 40 | 0.90^{1} = 0.90 |
36.00 |

Match 3 | 30 | 0.90^{2} = 0.81 |
24.30 |

Match 2 | 20 | 0.90^{3} = 0.729 |
14.58 |

Match 1 | 10 | 0.90^{4} = 0.656 |
6.56 |

Note that as you go back in time, each match performance is given 10% less credit (by multiplying by 0.9).

To work out the value of Atherton's weighted average, add up the last column values (W) and divide them by the third column's values (k):

131.44 / 4.095 = 32.1

In other words, Atherton's original average of 30 is boosted by the exponential average to 32.1 because his recent form has been better than 30.

If you do the same calculations for Hussain, you should end up with:

114.26 / 4.095 = 27.9

Here, Hussain's original average of 30 is reduced by the exponential average to 27.9, because his recent form has been worse than 30.

So this exponential gives more credit to the player who is in better recent form (Atherton in our example).

How much extra credit he gets depends on what value you choose for the decay factor. In the example above I chose 10%. But the actual figure chosen is based on trial and error to find an amount that gives a 'fair' result (whatever you decide that means). In the officially published ratings, the decay factor used is in fact 4%.

Try out what happens if you insert the following decay factors into the calculation:

- 0%
- 50%
- 100%

You should get the following results:

- Atherton 30, Hussain 30 (0% decay simply gives you the players' averages)
- Atherton 41.6, Hussain 18.4
- Atherton 50, Hussain 0 (100% decay gives you the players' scores in the most recent match!)