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If you have ever tried to explain the concept of Pythagorean Record to a baseball novice, you probably have had to answer the following criticism: “That counts the extra runs at the end of a blowout as much as other runs, even though it does not matter whether you win 10-0 or 15-0.” The answer that we give to that criticism is that teams that can take advantage of blowouts have better offenses and those type of teams will be more likely to win close games in the future. That is the reason that we have thousand-run estimators that try to approximate how many runs a team will score on average, and why we evaluate players with statistics like VORP-measured in runs over replacement player. Runs are the building blocks of wins, and you win by scoring more runs than your opponent. We cringe when we hear offenses evaluated by batting average because we know that the goal of offenses is to score runs, not get hits.

The Inning

However, with all of these run estimators that sabermetricians have developed, we often forget the context in which runs are scored-by innings. Teams get to score as many runs as they can before their opponents record three outs; then they get to try again eight more times. That environment-how much you can score before three outs-is the environment to keep in mind when we talk about winning games. Nearly a decade ago, Keith Woolner wrote about the link between runs per inning and runs per game, and how well you can predict the frequency of zero-run innings, one-run innings, two-run innings, etc. by looking at how many runs teams score per game.

It is certainly true that the rate of scoring a certain number of runs in an inning and the average number of runs per game are related. In fact, teams that have more variance in their run-scoring per inning also have more variance in their run-scoring per game. This is tricky to show because teams that score more runs also have more variance in the number of runs they score per game-that makes sense, because they have a lot of eight-run games, 10-run games, and 15-run games, so they are bound to have a higher variance because they needed enough big innings to put up those run totals. Simply checking the variance of runs per game against how frequently those teams have big innings would obviously yield a positive correlation. Instead, I needed some way to neutralize the variance of runs per game. I initially tried dividing by the number of runs per game, but that statistic still had a positive correlation with runs per game. I tweaked with things until I found a way to measure variance of runs per game that did not have any correlation (-0.0025) with runs per game, which I call “Adjusted Variance” or “AdjVar,” is this:

```          (Variance of Runs/Game)
((Runs/Game) ^1.30)
```

Looking at 1998-2008 data for each team (330 team seasons total), I found that this number was slightly positively correlated with the frequency of scoring zero runs in an inning (correlation = 0.097, two-sided p-stat = 0.075), highly correlated with the odds of scoring four or more runs in an inning (correlation = 0.258, two-sided p-stat = 0.000), and highly correlated with the odds of scoring five or more runs in an inning (correlation = 0.292, two-sided p-stat = 0.000). That much should not come as a surprise; we predicted that teams that have more variance in their runs-per-inning scoring would have more variance in their runs-per-game scoring.

Run-Scoring Variance and Pythagorean Record

The next step is to check if teams with more variance in their runs per game tend to underperform their Pythagorean records. In fact, this is true-the difference between actual wins and Pythagorean expected wins is negatively correlated with the AdjVar statistic above (correlation = -0.303, two-sided p-stat = 0.000). Teams that are more volatile in their rate of scoring runs are going to lose more often than other teams that score similar number of runs, but are not as volatile.

Now we know that teams that have high variance in their run-scoring by inning have more variance in their run-scoring per game. We also know that teams that have more variance in their run-scoring by game are not as likely to win as teams that put up the same number of runs but without as much of a spread. The next step is to figure out if there is any way to predict which offenses will have less variance in their runs per inning.

Which Offenses Spread Their Runs Around Better

Three years ago, Sal Baxamusa looked at 2006 team-scoring data and used the Weibull Distribution to predict how often they would score a certain number of runs. The Weibull Distribution does a pretty good job at predicting the number of times teams will put up certain run totals, but tends to underestimate how often teams are shut out. This is likely due to the fact that the talent level of pitchers is different, so analyzing how a team scores in general will not take this into account. You face Johan Santana sometimes, and you face Livan Hernandez at others, and Santana might shut you out more often than a model of hitting alone would predict. Baxamusa demonstrated that slugging teams were shut out less often, and also were more likely to score at least three runs in a game than their season run total and the Weibull Distribution would predict. This was useful information, but given the difficulties with the Weibull Distribution and the small sample size of just thirty data points, he was unable to check this in much detail.

By looking at runs per inning, we can look at a much larger sample-there were 477,884 half-innings from 1998-2008. Using this, we can check which type of offenses are more likely to spread their runs around and win more games as a result. The correlations between the odds of scoring at least a given number of runs in an inning and a number of common offensive rate statistics reveal even more evidence of Baxamusa’s suspicion-that the teams that score with power are more likely to win than other teams who score similar numbers of runs.

For reference, note that the average team from 1998-2008 only scored in 29 percent of the innings that they played, but they scored two or more 14 percent of the time, they scored three or more six percent of the time, they scored four or more three percent of the time, and they scored five or more one percent of the time.

Below I list the correlation between the frequencies of scoring at least a certain number of runs in an inning and on-base percentage and slugging percentage. Note that each of these have a 0.887 correlation with runs per game. You will notice an interesting trend:

```
At least
X Runs/inning	 OBP    SLG
1               .822   .872
2               .741   .723
3               .603   .573
4               .746   .716
5               .667   .611
```

The trend that you probably noticed is that high-slugging teams are more likely to pick up at least a run in an inning, but high-OBP teams are more likely to have big innings. The reason that this is so important is that we have shown that being able to spread your runs around different innings is more valuable than scoring a lot of runs in one inning, in terms of wins and losses, since high variance in run scoring tends to be correlated with underperforming your team’s Pythagorean Record. This means that all of our standard measures of run-scoring are overweighting the contribution of OBP towards winning and underestimating the contribution of SLG towards winning.

The connection can be highlighted even further by using regression analysis to predict the probability that a team scores at least X runs in an inning. I regressed the probability of scoring at least one, two, three, four, and five runs in an inning on on-base percentage and slugging percentage and found the following formulas:

```Prob(Scoring at least 1 run)  = -0.154 + 0.659*OBP + 0.526*SLG
Prob(Scoring at least 2 runs) = -0.224 + 0.686*OBP + 0.307*SLG
Prob(Scoring at least 3 runs) = -0.164 + 0.462*OBP + 0.171*SLG
Prob(Scoring at least 4 runs) = -0.090 + 0.235*OBP + 0.094*SLG
Prob(Scoring at least 5 runs) = -0.050 + 0.138*OBP + 0.039*SLG
```

The important thing to realize when looking at these formulas is that the coefficient on SLG gets smaller relative to the coefficient on OBP as you increase the number of runs per inning. Teams that string together a lot of baserunners are more likely to score by putting up big innings than teams that swing for the fences, who will spread their runs around better.

The link remains strong when you look at similar statistics for scoring at least a certain number of runs in a game:

```Prob(Scoring at least 1 run)   =  0.673 + 0.336*OBP + 0.387*SLG
Prob(Scoring at least 2 runs)  =  0.238 + 0.815*OBP + 0.814*SLG
Prob(Scoring at least 3 runs)  = -0.205 + 1.46 *OBP + 1.06 *SLG
Prob(Scoring at least 4 runs)  = -0.584 + 2.00 *OBP + 1.21 *SLG
Prob(Scoring at least 5 runs)  = -0.889 + 2.65 *OBP + 1.11 *SLG
Prob(Scoring at least 6 runs)  = -0.973 + 2.51 *OBP + 1.14 *SLG
Prob(Scoring at least 7 runs)  = -0.909 + 2.26 *OBP + 0.974*SLG
```

Conclusion

It is clear that power helps you score frequently, and on-base skill helps you pile on when you do score. In fact, a team’s home runs per at-bat has a 0.15 correlation with the difference between the number of wins a team gets beyond what their Pythagorean record predicts. Teams that hit more home runs do better than their Pythagorean Record suggests.

What this means is that power hitters are even more valuable than their VORP suggests. Power hitters not only change the scoreboard, but they change the scoreboard when it matters. The next time somebody tells you that a team is falling short because they rely too much on the long ball, you can reply that they may not rely on it enough.

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