# Ben Stokes, Joe Root and Square Roots.

## A mathematical subtlety buried in the ICC world cricket rankings

Cricketers are often put into three categories:  batsmen, bowlers and all-rounders.  All-rounders are players who are useful batsmen and bowlers.  England has a few of these, but the name Ben Stokes is the one that has been dominating the recent headlines.

How, mathematically, do you measure how good an all-rounder is?  Prepare to discover an obscure use of square roots.

In the ICC rankings, players are rated for batting and bowling using a weighted average, which is worked out on a scale between 0 and 1000, where 1,000 is the mark of perfect performance. (In reality, 900 points is usually the best that a great player ever reaches, while the greatest of them all, Don Bradman, peaked at 961 points in 1948).

On 29th May, before the start of the current Test match between England and New Zealand at Headingley, here were how two England players rated as batsmen and bowlers:

Batting points           Bowling points            TOTAL

Ben Stokes                 540                                320                      860

Joe Root                     872                               140                     1012

So who is the better all-rounder, Stokes or Root?  Root has more combined points, but that's not really an indication of all-rounded-ness.  If it were a matter of just adding up the points, a player with 1000 batting points who can't bowl would be equivalent to one with 500 batting and 500 bowling points.

What is needed is a measure in which a 500/500 player counts as a 500 all-rounder, whereas a 1000/0 player counts as a zero all-rounder.

The way to achieve this is simple: multiply together the batting and bowling points, and take the square root of the answer.

Batting points    Bowling points     Product    Square Root

100                               900               90,000          300

500                               500             250,000          500

1000                             0                             0              0

And that is how the ICC all-rounder points are calculated.  Actually in the official figures, the final step of taking the square root isn't taken because it doesn't affect the ranking order, but from a purist's point of view it should be.

So on May 29th Ben Stokes had an all-rounder rating of 540 x 320 = 172,800 (in the ICC table this number is indicated in thousands, so 173) while Joe Root had a rating of 872x140=122,080 (indicated as 122), which is why Stokes ranks higher than Root as an all-rounder despite having fewer batting+bowling points.

I wonder if at school, Ben Stokes ever asked his teacher: "Sir, when am I ever going to need square roots?"