# Details of bowling calculations

David Barry

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 Fig.1: A plot of runs remaining against balls remaining (click here for a bigger image) © David Barry Enlarge

Here's a plot of runs remaining against balls remaining, based on the FOW's (Fig 1). Note that only first innings were considered, since in the second innings teams bat to make a particular score, not to maximise their average total.

As you can see, the first few regression lines are all pretty close to each other.

To get a run-value of wickets based only on balls remaining, I went through each first innings wicket and found the run-value of that wicket and all subsequent wickets. Then for each wicket, I plotted that total run-value of wickets against the balls remaining after the first wicket (Fig. 2).

 Fig. 2: run value of wickets based on balls remaining (click here for a bigger image) © David Barry Enlarge

The regression line is forced through the origin.

Now let's move onto the short-term dip (Fig. 3). Plotted below is the difference in average run rate in the two overs after a wicket (compared to the overall average), at each over:

 Fig. 3: short-term run-rate dip after fall of a wicket (click here for a bigger image) © David Barry Enlarge

I've called it flat (with variations from random noise) until 15 overs before decreasing to zero, though there's room for debate there. The dip to zero makes cricketing sense.

The above gives us the run-value of wickets. Here is how I used them to tweak the economy rates.

The long-term credit (= 0.0277 * (balls remaining)) is either drowned out completely by the short-term credit, or includes it. If it is less in value than the short-term dip, then it gets completely ignored. If it is greater, then only the difference between the two is considered.

Now, each bowler bowls on average around 20% of the team's overs, so I gave the 80% of the long-term credit to the bowler. The remaining 20% would be incorporated into his actual economy rate anyway.

For the short-term credit, I gave the bowler 62.5%. The dip in scoring lasts about two overs. Let's say that on average, the wicket falls with three balls to go*. That leaves the next over (bowled by someone else), and three remaining balls. I guessed that the bowler has a 50% chance of bowling those balls (something to check), so half the time 50% of the dip benefits other bowlers, and the other half of the time it's 75%. Split the two, 62.5%. Plenty of work to sharpen those numbers is possible, but it shouldn't make too much difference to the overall results.

*But check this out. 122 wickets on ball 1 of the over, 131 on ball 2, 104 on ball 3, 104 on ball 4, 104 on ball 5, 124 on ball 6. Possibly random, possibly something there — perhaps batsmen take a ball or two to get their eye in against a new bowler.