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How to figure expected runs in an inning?
Posted: 20 April 2007 09:04 PM   [ Ignore ]

I was looking over’s run expectancy matrix and trying to apply it to the Phillies’ 4-2 win over the Nationals yesterday.

How would I figure how many runs the Phillies should have scored in the first inning, for instance?

Jimmy Rollins doubled to left (Fly). (RE: 0.53)

Shane Victorino singled to right (Liner). Jimmy Rollins scored. (RE: 1.16)

Shane Victorino advanced on a stolen base to 2B. (RE: 0.92)

Chase Utley flied out to right (Liner). Shane Victorino advanced to 3B. (RE: 1.16)

Pat Burrell walked. (RE: 0.96)

Greg Dobbs flied out to shortstop (Fly). (RE: 1.21)

Wes Helms reached on fielder’s choice to third (Grounder). Pat Burrell out at second. (RE: 0.52)

Would I add up the run expectancy of all 7 instances and divide it by 7? If so, would I exclude the RE of Victorino’s single and only include the RE following his stolen base?

Thanks for any clarification.

Posted: 21 April 2007 07:00 AM   [ Ignore ]   [ # 1 ]

The RE of a situation (that is, number of outs and men on base) is the average number of runs that we expect to score from that point until the inning is over. Looking at the second event of the inning:

Shane Victorino singled to right (Liner). Jimmy Rollins scored. (RE: 1.16)

The initial situation is man on 2nd, nobody out. According to BP’s run expectancy matrix from 2006, that has an RE of 1.15. Then Victorino singles, scoring Rollins. The final situation is man on 1st, nobody out, with a run scored. The RE for the rest of the inning is 0.93, again from BP’s 2006 data.

So first, we expect 1.15 runs to score. Then one run *does* score, and we’re in a situation where we expect 0.93 more runs to score. For his single, Victorino gets a credit of 1.93 - 1.15 = 0.78 runs above average.

As you can tell, the RE figure given in parentheses after each event is the RE just before the event takes place. (1.15 is roughly equal to 1.16, 0.93 is roughly equal to 0.92.)

The point of this step-by-step accounting is to give the right amount of credit to each batter. Reading from your list:

Rollins gets 1.16 - 0.53 runs above average for his double
Victorino gets 0.92 - 1.16 + 1 runs above average for his single, the +1 at the end for the run that scored
Victorino gets 1.16 - 0.92 runs above average for his SB
Utley gets 0.96 - 1.16 runs above average for his flyout (less costly than a usual out because the runner advances)
Burrell gets 1.21 - 0.96 runs above average for his walk
Dobbs gets 0.52 - 1.21 runs above average for his flyout (very costly when you don’t drive in the runner from 3rd with 1 out)
Helms gets 0 - 0.52 runs above average for his FC.

The total sum is 1 - 0.53 runs above average, since everything else cancels out. In other words: the average team scores 0.53 runs per inning, but the Phillies scored 1 run in that inning, which is 0.47 runs above average. If you add up the +/- figures for the six players involved, you get exactly 0.47.

You want to know how many runs the Phillies should have scored. We have the events of the inning, and the order they occurred in. Knowing both the events and the order, the number of runs that “should have” scored is exactly the number of runs that did score. Knowing just the events, but not the order they occurred, the question becomes: given a generic inning with a double, a single, a stolen base, a walk, and 3 outs—in some order—how many runs will score, on average? This isn’t a question that the run expectancy system is well designed to answer. Run expectancy tells us how much credit to give each batter if we know the order of events.

For your question (assuming I have interpreted it right) I recommend the BaseRuns formula:

Runs = (H + BB - HR) x [ B / (B + outs) ] + HR
B = .8 x 1B + 2.1 x 2B + 3.4 x 3B + 1.8 x HR + .1 x BB

This is taken from Tangotiger’s “How are runs really created” series (, which is essential reading. David Smyth developed the formula. Note that this is a simple version of the formula which doesn’t include SB/CS. If you want to add those in then you might use

B = .8 x 1B + 2.1 x 2B + 3.4 x 3B + 1.8 x HR + .1 x BB + .8 x SB - 1.2 x CS

No doubt the formula seems confusing and unmotivated. If you want explanation then read “How are runs really created.” Or just plug and chug.

In this inning, we have:
B = .8 x 1 + 2.1 x 1 + 3.4 x 0 + 1.8 x 0 + .1 x 1 + .8 x 1 - 1.2 x 0 = 3.8
outs = 3
H + BB - HR = 3

Therefore, estimated runs is given by
Runs = 3 x [ 3.8 / (3.8 + 3) ] + 0 = 1.7

So in a sense, the Phillies “should have scored” 1.7 runs. However, I think this is an overestimate. BaseRuns wasn’t designed to work for individual innings, so while it is reasonably accurate, it isn’t perfect. In particular I believe it will overestimate low-scoring innings (like this one) and underestimate big innings. Still I don’t know of any better formula.

Posted: 22 April 2007 08:57 PM   [ Ignore ]   [ # 2 ]

Thanks for all the information.

I was going to try to use RE to see how many runs the Phillies have generally missed out on, but I’ll look into BaseRuns as well.