Not All One-Run Games are Created Equal

It’s the bottom of the fourth. No outs. Your beloved Milwaukee Brewers are up to bat trailing the Dodgers 1-0, with Clayton Kershaw on the mound. They’ve picked up two scattered hits and drawn a walk over four innings, but the sentiment in the dugout and the stands seems to read if they haven’t scored yet, chances don’t look so good.

Consider the same situation, now, with one small change. Your Brewers are still down by a run. It’s still the bottom of the fourth. Kershaw is still dealing. But it’s 2-1 Los Angeles this time. Milwaukee has still only gotten two hits and drawn a single walk, but the timing has worked out such that a run scored. By the numbers, things are almost exactly the same. No question about it. The sentiment, though, is certainly different. We’ve broken through once already, think the players, manager, and fans. We can do it again. Well, of course the Brewers can do it again. But, statistically speaking, will they? That is: when trailing by one run as they enter a half-inning, is a team more likely to come back in a non-shutout than in a game in which they haven’t yet scored?

The answer is “yes,” although only by what initially appears to be a small margin. In 2013, 5705 half-innings began with the batting team trailing by a run. 11.4% (651) of those half-innings ended with the batting team tied or in the lead. The same year, 2915 half-innings began with the batting team trailing specifically by the score of 1 to 0. 11.1% (324) of those ended in a lead change or tie.

At first glance, a 0.3% difference between odds of scoring when down by a run versus the specific case of being down 1-0 seems minor. And it is, really. For years with complete-season data available since 1871, the percent of half-innings started where it’s a one-run game and the losing team up to bat which resulted in a lead change or tie (let’s call this %ORLC) averages out to 11.5% ± 1.3% (1 σ). The subset of these in which the batting team was being shutout (let’s call this %ORSLC) has an average of 10.6% ± 1.1% (1 σ). Middle-school statistics will tell you that while, yes, %ORSLC is on average nearly a percent lower than %ORLC, they fall within a standard deviation of each other and, thus, their difference is not statistically significant.

That’s true. But baseball isn’t middle-school statistics and two subsets whose error ranges overlap are not for all practical purposes equal. Quite remarkably, %ORLC has exceeded %ORSLC for each consecutive season of Major League Baseball since 1977 (when %ORSLC was 0.2% higher) and every year since 1871 except for five seasons (out of the 111 years of complete-season data that were available).

That is: in 106 out of the last 111 seasons for which box scores have been logged every game, a batting team behind in a one-run ballgame has successfully erased the deficit more often when not trailing 1-0. The margin isn’t huge, of course, but the trend is meaningful.

Above: Percentage of one-run game situations and specific 1-0 game situations (%ORLC and %ORSLC, respectively) in which the team losing scores to tie or take the lead

After all, baseball is a game of small but meaningful margins. The 111-year average relative difference between these two metrics (10.6% vs 11.5%) is proportional to a .277 batting average versus .300, or 89 wins in a 162-game season instead of 97. The latter is perhaps a more relevant comparison, since it is gaining (and maintaining) a lead that is crucial to winning games.

Among teams in 2013, however, these differences aren’t so marginal. In %ORLC (percentage of half-innings in which a team trailing by a run ties it up or takes the lead) the Royals finished first at 16.7% and the Cubs finished last at 6.5%. In %ORSLC (same stat but for the score 1-0), the Rays finished first at 16.7% (same number, coincidentally) and the Red Sox finished last at 4.9%. Considering the Royals didn’t make the playoffs in 2013 and the Red Sox won the World Series, I wouldn’t use %ORLC and %ORSLC as indicators of a team’s ultimate success unless you’re looking to lose a lot of money in Vegas.

While one could theorize for hours on the meaning and utility of each made-up statistic, it sure doesn’t seem like %ORLC and %ORSLC are indicative of much on a team-by-team basis. But that doesn’t mean they’re useless. Let’s go back to the long-term trend of %ORLC and %ORSLC, where the former was higher than the latter 106 out of 111 times.

Some underlying process, it would seem, must be responsible for this impressive stat. If we are to believe that teams truly underperform, ever so slightly, when they’re losing 1-0 due only to the fact that they’re being shut out, shouldn’t we able to see the effect of psychology on performance somewhere else?

As it turns out, you don’t have to look far. Let’s consider the general situation of a team coming up to bat down by a run (not only the specifically 1-0 case), which is colloquially termed a “one-run game.” We’ll abbreviate any instance of this (a trailing team coming to bat in any half-inning) as OR. Now this situation could happen at any point in a game. A visiting team leads off with a run in the top of the 1st, the home team comes up to bat – that’s an OR. It’s all tied-up in the top of the 13th, the third baseman slugs a solo shot to left, three outs are recorded, the home team steps up the plate with one chance to stay alive – that’s an OR. So, in what inning on average does an OR occur?

In 2013, the answer was the 4.95th inning. In 2012 and also for the last 111 years of available records, the 4.91st inning. Baseball amazes us once again with its year-to-year consistency in obscure statistics. But this obscure stat isn’t all that meaningful on its own. Okay, so most one-run situations occur near the 5th inning – so what?

Well, let’s take a look now at the average inning in which a team scored in an OR to tie or take the lead. We’ll call this a one-run game situation where the lead changes, or ORLC. In 2013, of all the instances of ORLCs, the average time they occurred was the 5.18th inning. In 2012, the 5.10th inning. And for the same 111 seasons of recorded game data, the 5.20th inning. Once again, we see a marginal but nonetheless compelling deviation from the average, just as we saw with %ORSLC. Teams score in one-run situations about a third of an inning later than the one-run situations tend to occur themselves. That may not seem like a whole lot, but consider that in our 111-season dataset only two years – 1902 and 1912 – saw earlier ORLCs than ORs on average. Just two years in one-hundred eleven.

Above: Average innings of occurrence for one-run game situations (OR) and one-run game situations in which the trailing team scores to tie or take the lead (ORLC)

So what’s going on? I like to think of average ORLC minus average OR as a league-wide statistic for urgency. Consider the following: if the inning number had no effect on the performance of a trailing team in a one-run situation, then we would see roughly the same average inning of occurrence for both OR and ORLC. Out of 111 years, we’d expect to see about 55 years in which OR occurred earlier on average than ORLC and around 55 in which it didn’t. But we don’t see this at all, which strongly suggests that inning number has an effect on how a team does at the plate when down by a run. This is the urgency statistic. It describes a trend that has rung true for the past 101 consecutive seasons of Major League Baseball – when time is running out and the 9th inning is rapidly approaching, teams in close games get their acts together and produce runs. Not every time, of course, but we’re speaking in averages of massive sample sizes here.

So, while your Brewers are likely to fare worse trailing Kershaw and the Dodgers 1-0 than 2-1, take solace in the fact that it’s the fourth inning. Statistically speaking, they’ll have a better chance breaking through as the game goes on and their need for a run becomes more urgent. The effect of team psychology has left its imprint on the records of baseball games since the sport’s earliest days.

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I wonder how much of this has to do with pitchers tiring later in games.


This is really what I was thinking. Someone on here wrote an article recently about comparing a bunch of bullpen arms that only throw 1 inning each vs someone like King Felix in the 1 game wildcard playoff. They showed that after about 3 innings or so opponent batting average against was significantly lower for the relievers just because they had fresher arms than the staff “ace”.


Is it really psychology, or just the likelihood that the pitchers in the 1-0 are better than average, which would decrease the odds of overcoming the one-run deficit?

If you did the comparisons one pitcher at a time, so for example you compared trailing Kershaw 1-0 to trailing Kershaw by one run when you’ve scored at least once, then you might have something on the psychology angle.

Sean Dolinar

This looks a lot like selection bias. A team who hasn’t score is less likely to score than a team that already has. I also think that scoreless innings and shut out games are underestimated if you used a purely random Poisson distribution. My guess is the underlying mechanism would be the disparity in teams.


I guess most of this has been brought up, but:

Pitchers that have 1-0 are generally better pitchers than 4-3 pitchers. Pitchers that have a 1-0 lead have generally thrown fewer pitchers than those in 3-2 leads. Overall *pitching environment* will also affect the year-over-year. Pitchers effectiveness decreases through each line-up turn as well.

Whenever we assert a non-statistical premise, the only way to prove it is basically eliminating all statistical possibilities; which, kinda sucks.

Elliot F

There’s a much simpler explanation than psychology, pitcher fatigue, or anything else: In the first place, offenses are better in the late innings for the simple reason that the pitcher is more likely to be removed for a pinch hitter (in the NL and in all of MLB until the early 1970s). A pitcher who is losing 1-0 is less likely to be removed for a PH in a high-leverage situation than a pitcher losing 2-1, and so on down the line. That explains the bias against 1-0 and also explains why the later innings are more productive offensively. Another… Read more »