An analysis of the Duckworth/Lewis System

The Duckworth/Lewis system is the system now used to determine the winning score in rain-interrupted one day matches. The system was updated in September 2002 to take account of the higher scoring in recent times. For example, the average 50 over score has been increased from 225 to 235.

Click here to see the 2002 Duckworth/Lewis table.

The system was changed again in October 2003 in order to work better for matches with high scores. This adjustment requires a computer program to calculate, and is known as the Professional Edition. Using the 2002 tables above is known as the Standard Edition and is used for matches below international level. The analysis below uses the Standard Edition of the Duckworth-Lewis method.

This page analyses the accuracy of the system, using 650 one day internationals played between 2 January 2001 and 28 May 2006. The over by over scores were collected for matches where the team did not have its maximum number of overs reduced due to interruptions by rain or due to slow over rates. For each over, we compare the final score against the expected score by Duckworth/Lewis. In the case of a team batting second achieving the winning target with balls to spare, a final score is estimated by applying the D/L system to the team's winning score.

The D/L system converts the number of overs remaining and the number of wickets lost into a "resources remaining" figure. At the start of the innings, this is 100, but as overs are completed or wickets fall the "resources remaining" falls. The expected score is the current score plus 235 times the resources remaining divided by 100. The system assumes that the average innings score for an ODI is 235.

The formula used by the D/L system is:

Z(u, w) = Z0(w)[1 - exp{-b(w)u}]

where Z(u, w) is the expected number of runs to be scored in u overs when w wickets have been lost. Z0(w) is the average total score if an unlimited number of overs were available and when w wickets have been lost. b(w) is a decay constant that varies with w, the number of wickets lost.

Duckworth and Lewis do not list the values of the constants Z0(w) and b(w). The following table lists values based on fitting the above formula to the actual over by over scores in the one-day internationals mentioned above. The values are calculated separately for the side batting first and the side batting second.

For example, suppose that a team had lost 2 wickets after 10 overs. The table estimates that a team batting first would score 186 runs in the remainder of the innings. It estimates that the team batting second would score 178 more runs. For comparison, the D/L method estimates 183 more runs (the pre-2002 D/L method estimates 175 more runs).

Team batting first Team batting second
Wickets lost (w) Z0(w) b(w) Z0(w) b(w)
0 297.16 .034005 560.83 .010787
1 252.09 .040588 469.11 .012625
2 219.99 .046460 386.49 .015409
3 180.47 .057669 294.09 .020311
4 144.83 .074333 201.23 .030664
5 108.16 .092476 136.01 .044826
6 85.494 .12285 109.01 .044826
7 49.176 .22613 67.961 .064166
8 28.113 .32446 42.509 .10788
9 13.027 .56227 18.205 .14941

The following table lists the number of runs difference between the D/L score and the actual score, the standard deviation of the difference between the D/L and actual scores and the number of innings that reached that number of overs. The table also includes the difference that would be obtained by using the pre-2002 D/L table. It is clear that the Standard Edition of the D/L method is reasonably accurate, with the team batting first scoring slightly more than the table predicts and the team batting second scoring slightly less than the table predicts.

Note that in the first table entry we can see that the average innings by a team batting first was 238.7 runs (235 - (-3.7)) and the teams batting second averaged 232.4 runs (235 - 2.6).

Team batting first Team batting second
Overs
Completed
D/L minus
actual
Standard
Deviation
Pre-2002
D/L minus
actual
Innings D/L minus
actual
Standard
Deviation
Pre-2002
D/L minus
actual
Innings
0 -3.7 60.3 -13.7 650 2.6 57.0 -7.4 599
5 -5.5 54.5 -14.4 650 1.6 50.7 -7.2 597
10 -6.0 48.4 -13.5 650 2.5 44.7 -5.2 595
15 -5.6 42.2 -11.2 650 4.1 39.5 -2.4 590
20 -6.2 37.3 -10.0 647 4.7 35.3 -0.5 575
25 -6.9 32.3 -8.9 645 5.8 32.3 1.8 552
30 -5.6 28.6 -6.1 645 5.4 26.8 2.6 528
35 -6.5 24.2 -5.8 631 4.9 23.3 3.2 492
40 -7.0 19.8 -5.5 615 3.5 18.3 2.7 441
45 -5.7 12.9 -4.2 587 2.0 11.3 1.9 345
46 -4.8 11.4 -3.4 579 1.8 10.0 1.8 303
47 -3.9 9.4 -2.7 571 1.5 8.3 1.6 270
48 -2.7 7.3 -1.8 555 0.9 6.7 1.0 216
49 -1.6 5.0 -1.2 524 0.8 4.3 0.9 147


Please email any comments or queries to Shane Booth.

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Last modified: August 14, 2006