Friday, April 20, 2012

On Why A Century Is Not Entirely An Arbitrary Milestone

The figure 100 is a nice round figure. It has been under scrutiny recently especially because Sachin Tendulkar reached his 100th international century. But why does it matter? Does a player who makes 95 contribute significantly less to a team than a player who makes 105? Is the difference between scoring 95 and 105 significantly dissimilar to that between scoring 85 and 95?

Milestones get established because thresholds matter. These thresholds are apparently arbitary. It is probably fair to say that they are in fact arbitrary, in the sense that they were not established for a cricketing reason.

It turns out that 100 runs is not such a bad threshold score after all. My friend Amogh suggested this idea to me a couple of weeks ago and did some calculations. His idea was based on the equivalence between a century by a batsman and a 5 wicket haul by a bowler. A 5 wicket haul has some correlation to the total number of wickets required to bowl a side out. Here is how the argument goes.

Assuming that a team is made up of 4 players whose primary job is to take wickets - "bowlers" and 7 players whose primary job is to score runs - "batsmen", it follows that a 5 wicket haul is double the number of wicket each bowler can be expected to take on average in Test Team (10/4 = 2.5 etc.). Hence, a 5 wicket haul, even though it is probably just as arbitrary as a century, as a threshold to be recorded, is meaningful beyond its putative gestalt. It accounts for the number of times that a single bowler dismissed at least half the opposition in an innings.

How should a comparable measure be established for batsmen?

We tried this in a number of ways. Declarations create difficulties, especially in the third innings. Batting in the second innings (3rd or 4th match innings) creates difficulties in general, because of the existence of preset targets and declarations.

As a comparable achievement to the 5 wicket haul, the following criteria are used -

The minimum run total in the 1st innings (1st or 2nd match innings) for the loss of 10 wickets, which ensures that a team is more likely to either win or draw the Test, than it is to lose it.

The minimum run total is calculated by calculating the runs/wicket figure for each 1st innings (1st or 2nd match innings) total, and sorting the runs/wicket score in ascending order. The threshold team score is that (runs/wicket x 10) figure at which the likelyhood of defeat is less that 50%.

If this team total is N, then the "century" score is 2*N/7 (Since we assume that there are 7 batsmen who are primarily responsible for scoring runs, and, as in the case of the 5 wicket haul, the threshold must be double the par score.)

The table below shows the development of the Threshold Team Score and the "Century" Score at 200 first innings intervals (roughly every 100 Test Match). The figures are cumulative - the figure for the 1000 innings mark includes the first 1000 innings, the figure for the 2000 innings mark includes the first 2000 innings and so on.

The "Century" scores have ranged between 88 and 99 over these years. This has remained remarkably stable over the 130 odd years that Test cricket has been played. The high points (where the "century" score has been 99), were when Bradman played, and in the 2000s.

Why consider only 1st innings scores?
This is because it is difficult to consider 3rd or 4th innings scores due to the existence of leads, deficits, declarations and targets. While teams often declare their 1st or 2nd innings closed, this is a different type of declaration after a different type of innings compared to the 3rd or 4th innings. If the proposed calculation is done for 2nd innings, then a "century" score turns out to be about 160, because a large share of 2nd innings are played with a large lead, involve 3rd innings declarations, can involve very short 3rd or 4th innings, etc.

Please note that this is an attempt to look at a century as a statistic. This does not mean that a particular score of 88 in a 1992, was as good as a particular score of 99 in 2012.

What this does show, is that the threshold of 100 is not a bad one, and the comparison of a century to a 5 wicket haul is a good one.


  1. Such an ingenious analysis, with a very satisfying result. Could you be persuaded to pursue this line to provide scores that lead to other result probabilities (say 50% chance of winning; 90% chance of avoiding defeat)?

    Have you come across the academic work of Dr Scarf on predicting Test match outcomes? I've referenced it on Declaration Game. I'll send you a link if interested.

    1. Great analysis....
      Please send the link to me as well...

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    2. The academic work:
      An analysis of strategy in the first three innings in test cricket: declaration and the follow-on. Philip Scarf and Sohail Akhtar. Salford Business School Working Paper Series. Paper no. 337/10.

      And where I've used it:

  2. Great analysis KD. Here's one more proof why it may not be such an arbitrary number:

    Look at the graph and the cut-off value.....


  3. In my personal view i don't think that a century can express a player's expertise however, consistence of performance and stamina express a player's ability.

    1. You are right, any one can score a century.. but if he consistently scores them then we can say he is a class player.
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