Yeh Dil Maange More? - Part 1

A few nights ago, Yusuf Pathan was making his way through the Kolkata Knight Riders' inadequate bowling attack. On one particular delivery, he struck the ball so hard, that it rose high for a few seconds before landing in a no-man's land area.

At this, my mother exclaimed "Had there been a fielder standing right there, he would have been out!"

This kick-started a chain of thoughts. What if there were 15 players in a team? Would the average score of a team playing a Test, an ODI or a T20 be higher or lower? What if there were 5 players?

So,I set about solving this problem and thanks to the extensive data available at http://www.cricinfo.com, I have been able to dig out some much needed information. Here's an attempt.(I would really want to develop this exercise and would need your comments on improving this further.)

Let us take a few assumptions before proceeding any further.
1) I am only considering a One-Day International only, with 50 overs a side.
2) For the ease of data collection, I have included data from all countries playing ODIs (recognized by the ICC)-Countries from Australia to UAE.
3) Data from all past matches have been included, including the ODIs that were abandoned or those which produced no results.
4) For the purpose of solving this problem, I shall proceed assuming that there are more than 11 players in a team.


Facts governing the course of scoring in the match

1) More the number of players, more the number of lesser quality batsmen and bowlers.
2) 6's will not be affected, 4's and other runs will be affected.
3) Batsmen coming in the end are generally coming in towards the end of the team's innings, so they hit hard and get out. This assumption stays true for the newly added batsmen too.


Batting Averages for each player against any opposition:

Player 1: 32.34
Player 2: 31.90
Player 3: 32.81
Player 4: 34.70
Player 5: 30.50
Player 6: 26.36
Player 7: 21.07
Player 8: 16.63
Player 9: 13.15
Player 10: 10.51
Player 11: 07.54


On the curve for the averages itself, I have superimposed a 6th degree function which approximates the curve the best. The function is-
y = 0.0006x6 - 0.0264x5 + 0.4687x4 - 3.9523x3 + 15.62x2 - 26.261x + 46.544
With the R2 (Remember statistics?) value as-
R2 = 0.9981

The significance of the value of R2 is such that the closer the value of R2 is to 1, the better fit the curve is to the original data.

I have extended the same curve using the values of the averages from batsmen from position 8-11 to extrapolate these values for the new hypothetical players using a quadratic function. I used a quadratic function because it seemed to fit the curve the best possible way and gave non-negative values for the averages for the hypothetical players.



The averages for the new hypothetical players has been found to be-


Player 8: 16.63
Player 9: 13.15
Player 10: 10.51
Player 11: 07.54
Player 12: 04.57
Player 13: 02.4212
Player 14: 01.915
Player 15: 00.2376

The new function used is -

y = 0.2063x2 - 4.2051x + 20.801
And the new R2 value is-
R2 = 0.9948

Result #1
Therefore, extrapolating just the averages of each batsman according to the rules we laid down shows that the Average of the Total Score of a Team of 15 rises by 3.55% only.


But merely just adding the averages for the 15 players will not give us our answer. We need to consider strike rates and run rates too.

Watch this space...