Pappus' plane - cricket stats

This entry is inspired from a line from The Best of the Best. On Hobbs and Sutcliffe, Charles Davis writes that, "[e]ach was a great batsman in his own right, but even that is not quite enough to account for their performances together".

Given the individual averages of two openers, how much would we expect their partnerships to average? And which opening pairs do the "most better" that you would expect?

To answer these questions, I took all opening pairs who opened the batting at least 15 times together. I ordered each pair so that the first had the lower average of the two (so that, in the tables and equations below, avg1 is the lower individual average, and avg2 is the higher). Common sense suggests that the average partnership should be more determined by the lower individual average, since that batsman is more likely to get out first.

Note that I've used individual averages as openers when doing this analysis.

I then threw the data into gretl, an econometrics program. Since there are two independent variables (one for each opening batsman), I can't easily make a pretty graph. You'll just have to cope with equations and tables. Here is some of the output:

Modèle 1: Estimation en MCO avec 97 observations 1-97
Variable dépendante: avg_part

      VARIABLE       COEFFICIENT        ERR. STD         T           p. critique
  const                -7,60135          5,30561      -1,433   0,15526
  avg1                  0,484575         0,144117      3,362   0,00112 ***
  avg2                  0,766951         0,157437      4,871  <0,00001 ***

  Moyenne de la variable dépendante = 41,6219
  Écart-type de la var. dép. = 12,6076
  Somme des carrés des résidus = 7632,06
  Erreur standard des résidus = 9,01067
  R2 non-ajusté = 0,499844

You'll note that my computer is French. The word moyenne is 'mean', écart-type is 'standard deviation', and the other words are close to their English counterparts. If you don't know what they mean, that is not important.

The table tells us that, "on average", we expect that the average opening partnership (avg_part) should obey the following equation:

avg_part = 0,484575*avg1 + 0,766951*avg2 - 7,60135.

The R² value says that roughly half of the variance in the data-set is explained by this model.

Obviously the equation isn't valid everywhere — if both openers average zero, you would not expect them to score negative runs! But roughly 47 of the 97 opening pairs in the sample do better than the equation, and 50 do worse, so it appears to be pretty much "in the middle".

It is surprising (to me, at least) that the co-efficient of avg2 is so much higher than that of avg1. This says that it is the opener with the higher average who more determines the size of the average partnership. I'm at a bit of a loss to explain this. Perhaps openers with lower averages have lower strike rates (so while they don't score as many runs, they don't get out first)?

Now we get onto the pairs who do better than they should. In the following table, I've given the individual averages-as-openers, the runs scored together, the number of partnerships, 'obs' the observed average partnership, 'exp' the expected average partnership based on the equation above, and the ratio of the observed to expected.

opener1         opener2       avg1    avg2    runs  inns  obs     exp     ratio
T Franklin      J Wright      23,00   38,12   1543  28    55,11   32,78   1,68
Javed Omar      Nafees Iqbal  22,08   25,60   665   19    35,00   22,73   1,54
P Roy           V Mankad      31,71   40,74   868   16    57,87   39,01   1,48
J Stollmeyer    A Rae         41,94   46,18   1349  21    71,00   48,14   1,47
B Murray        G Dowling     23,92   31,55   786   20    39,30   28,19   1,39
C Cowdrey       G Pullar      42,42   43,84   906   15    64,71   46,58   1,39
C McDonald      A Morris      39,40   45,69   949   15    63,27   46,53   1,36
J Hobbs         H Sutcliffe   56,37   61,11   3249  38    87,81   66,58   1,32
Imran Farhat    Taufeeq Umar  33,10   39,30   754   15    50,27   38,58   1,30
Sadiq Mohammad  Majid Khan    34,93   42,23   1391  26    53,50   41,71   1,28

And it's a Kiwi pair who finish first! I suppose that if you analyse enough Test data, you'll eventually find New Zealand coming first in something.

My guess that openers with lower averages score slower certainly applies to Trevor Franklin, who is the fourth-slowest batsman of all-time according to Davis's list (qual. 1000 runs or 2000 balls faced; average over 20).

It may just be coincidence that some opening pairs do well in that table — perhaps they both had a good run of innings while batting together, or maybe they batted against weaker teams (I haven't tried adjusting for strength of bowling attack). But it may also be that they bring out the best in each other. Or, as Davis suggests in the case of Hobbs and Sutcliffe, that they held a psychological edge over their opponents when together.

When Stuart sees the other end of the table, he will be happy to see Graeme Wood coming dead last.

opener1         opener2       avg1    avg2    runs  inns  obs     exp     ratio
M Elliott       M Taylor      35,32   43,50   721   23    31,35   42,88   0,73
M Dekker        G Flower      15,86   29,30   357   22    16,23   22,56   0,72
B Woodfull      B Ponsford    50,90   54,18   860   22    40,95   58,62   0,70
G Gooch         T Robinson    43,88   44,97   621   19    32,68   48,15   0,68
R Simpson       L Hutton      25,92   56,48   477   15    31,80   48,28   0,66
E McMorris      C Hunte       26,86   45,07   548   21    26,10   39,98   0,65
B Pocock        B Young       22,93   32,13   378   21    18,00   28,15   0,64
Wasim Jaffer    V Sehwag      35,82   51,29   619   21    29,48   49,09   0,60
Hannan Sarkar   Javed Omar    20,66   22,08   207   18    11,50   19,34   0,59
A Hilditch      G Wood        31,56   33,61   354   18    19,67   33,47   0,59

It is also interesting that Javed Omar comes both second and second-last. Mark Dekker has easily the worst average of any opening batsman who's opened the innings 15 times.

For all the hugging, Langer and Hayden did slightly worse than would be expected, with a ratio of 0,92. Their average opening partnership of 52,08 is quite good (22nd on the list), but they each have excellent individual opening averages (48,94 and 52,66). The Langer/Hayden and Boycott/Amiss pairs are the only ones to have an average partnership of over 50 and a ratio below 1.


Figures are based on Tests 1 to 1858, that is up to the first Test between New Zealand and Bangladesh.