Pappus' plane - cricket stats

Captaincy is one area of cricket that does not receive much statistical scrutiny. It is not hard to figure out why — about the only thing you could easily compute would be how many wins the team had, and this figure is strongly dependent on the quality of the team. To remedy this, I've come up with a way of estimating how many wins, draws, and losses that a captain would be expected to have, given the strength of his side, the strength of the opponents, and whether they're at home or away. Once we have these, we can compare to the captain's actual record, and see who does better or worse.

First, I'll explain how to get the expected results. This method is a bit rough and can certainly be improved in places, but overall I think it does a good job. For each team, I calculate the average batting average (averages weighted by the averages of the bowlers faced), and average bowling average (with wickets wickets weighted by the batting average of the batsman dismissed). The latter is a little bit tricky — some teams use more bowlers than others. So, for each innings, I weighted the bowling averages by the number of balls bowled by each bowler. Then, if two innings were bowled, I took the average of the two.

Then you subtract the average bowling average from the average batting average, and you get a rating for the team. Do the same for the other side, and you get a measure of the difference in strength between the two sides.

Next you go through all Tests, calculate the difference in strength (to make things consistent, I did home team rating minus away team rating), and find how many wins, draws, and losses there are at various differences in strengths. I did this by binning all Tests into 20 bins. Plotted on the graph below are the expected fraction of wins and draws.



The fractions of wins does basically what we'd expect — it starts out flat and very low for teams that are outclassed, before rising steadily before plateauing. There are always going to be some draws (because of rain), so the fraction of wins won't hit zero or one. Even the weakest of home teams can achieve a draw rate of about 30% (well, maybe not Bangladesh), whereas very weak teams away can only draw about 20% of Tests.

The trend in draws is a bit different. It seems to go gently upwards until the teams are evenly matched, and then more sharply downwards as the home team becomes stronger.

I approximated these curves with piecewise linear functions. For the draws, it's flat for x less than -27, then upwards so that it hits the y-axis at y = 0,424, then downwards until x = 17, and then flat, at a value of 0,185.

For the wins, it's flat at 0,031 below x = -13,7, then upwards until x = 17,2, and then flat at a value of 0,785.

So now, for each Test, I calculate the difference in strength. Then I plug that number into the fitted graphs to get a fraction of a win, draw, and loss. For example, suppose that the teams are evenly matched. Then the home side gets 0,366 wins; 0,424 draws; 0,21 losses. The wins and losses for the away side are flipped: 0,21 wins and 0,366 losses.

You do this for each Test that a captain plays, and add up the expected wins, draws, and losses. Now we can compare to the actual record.

There's a question here about how to deal with draws. I decided to ignore them, for a couple of reasons. The first is that teams which score runs faster should have less draws, but I didn't take strike rate into account when doing the regressions above (I don't have strike rate data for all Test batsmen). Also, all Tests in Australia (as well as some elsewhere) were played to a finish between 1882/3 and World War II — no draws in a major cricketing country for over sixty years!

So instead I calculated the fraction of wins out of matches that ended in a result, that is: wins / (wins + losses). Do this for the actual value, divide by the expected value, and you get a ratio saying how much better or worse the captain's record is compared to what would be expected.

Whether or not it is reasonable to ascribe all the difference to the captain is certainly debatable, but it seems the best thing to do for now. Here are the best captains, as measured by this statistic. Qualification 20 Tests as captain.

                    ----expected----  --actual--  exp   act
name            mat w     d     l     w   d   l   w%    w%    ratio
Abdul Kardar    23  5,1   7,1   10,8  6   11  6   0,32  0,50  1,56
GP Howarth      30  7,9   11,1  11,1  11  12  7   0,42  0,61  1,47
J Darling       21  5,8   7,8   7,4   7   10  4   0,44  0,64  1,44
JM Brearley     31  11,3  12,0  7,7   18  9   4   0,60  0,82  1,37
Inzamam-ul-Haq  33  6,8   11,9  14,3  10  10  13  0,32  0,43  1,35
MP Vaughan      41  15,0  14,5  11,5  21  11  9   0,57  0,70  1,24
RB Richardson   24  8,4   8,2   7,4   11  7   6   0,53  0,65  1,21
GA Gooch        34  8,1   12,6  13,3  10  12  12  0,38  0,45  1,20
CA Walsh        22  5,7   7,6   8,7   6   9   7   0,40  0,46  1,16
DG Bradman      24  11,7  7,7   4,6   15  6   3   0,72  0,83  1,16
SP Fleming      80  23,6  27,1  29,3  28  25  27  0,45  0,51  1,14
RB Simpson      39  10,9  14,2  13,9  12  15  12  0,44  0,50  1,14
IVA Richards    50  21,7  18,2  10,1  27  15  8   0,68  0,77  1,13
CH Lloyd        74  31,3  27,0  15,7  36  26  12  0,67  0,75  1,13
N Hussain       45  14,0  15,5  15,5  17  13  15  0,48  0,53  1,12

Abdul Kardar, Pakistan's first Test captain, comes out on top. You can see that he didn't actually win many Tests, but his team managed to draw a lot that they "should" have lost. It is reassuring to see Mike Brearley so high up. Inzamam and Michael Vaughan are fifth and sixth.

Most of those high in the table did not have extended careers as captain, so perhaps some of them were just lucky and are higher than they should be. Of those with at least 50 Tests, Stephen Fleming is the best, just shading Viv Richards and Clive Lloyd.

At the other end, we have those who took their teams and un-inspired them to ineptitude:

                    ----expected----  --actual--  exp   act
name            mat w     d     l     w   d   l   w%    w%    ratio
M Azharuddin    47  18,5  17,2  11,3  14  19  14  0,62  0,50  0,80
HH Streak       21  5,0   6,1   9,9   4   6   11  0,33  0,27  0,80
MW Gatting      23  5,1   8,9   9,1   2   16  5   0,36  0,29  0,79
CL Hooper       22  5,2   7,8   8,9   4   7   11  0,37  0,27  0,72
BS Bedi         22  7,0   7,8   7,2   6   5   11  0,49  0,35  0,72
DI Gower        32  7,0   12,0  13,0  5   9   18  0,35  0,22  0,62
JR Reid         34  5,3   11,1  17,6  3   13  18  0,23  0,14  0,62
AC MacLaren     22  6,6   7,9   7,5   4   7   11  0,47  0,27  0,57
KJ Hughes       28  7,3   10,3  10,4  4   11  13  0,41  0,24  0,57
A Flower        20  3,5   6,4   10,1  1   9   10  0,26  0,09  0,36

Kim Hughes was pretty lucky to get to captain Australia in 28 Test matches.

A couple of others to finish (feel free to request others).

Mark Taylor 0,93, but probably disproportionately many of his losses were in dead rubbers. He also only had 11 draws (expected 17,5), in the era before Waugh made draws almost extinct for Australia.

Sunil Gavaskar 1,10 is the best Indian captain. He achieved this record by turning five expected wins into draws, and seven expected losses into draws. Not exciting stuff, but it gave him an overall positive record (9 wins, 30 draws, 8 losses).

Imran Khan at 1,08 didn't do that much better than he should have, and was also pretty drawish (14-26-8, expected 17,6-18,0-12,3). He did a bit better than Javed, who scores 0,99.