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Baseball’s balance in math

Earlier today, I wrote a little piece about what I called the balance that baseball is finding with all the strikeouts and all the home runs. I will have an interesting follow (I think) tomorrow as Bill James weighs in.

But before doing that, I wanted to share a little cool math from Tom Tango.

In the piece, I point out that teams in 2017 are essentially scoring the same number of runs as teams did in 1993 … but in VERY different ways. In 1993, hitters did EVERYTHING better except hit home runs. They hit for a significantly higher average, walked more, stole more bases, struck out a ton less often and averaged 4.6 runs per game.

In 2017, with a bunch more home runs, teams are also averaging 4.6 runs per game.

Well, Tango can show us how it works in math. If you like math, I think you’ll get a kick out of this. If you don’t — yeah, you can pretty much stop here.

In 1993, hitters had .80 more singles per game. They had a few more triples, a couple less doubles, so essentially the difference comes down to .80 more singles per game.

Using linear weights, which gives a value to everything, a single is worth .46 runs.

So you simple multiply .80 x .45 and you get: 0.37 runs. THat’s how many runs per game hitters today have to make up because of their lack of singles. Remember that number.

Well, hitters today are hitting .34 more home runs per game, which is a lot. A home run’s linear weight is worth 1.40 runs so .34 * 1.40 = .48 runs. So that’s quite a bit more than the singles value.

So why aren’t teams scoring MORE runs than in 1993? Ah, that’s the cool part of this — because if one group of hitters is hitting .80 more hits per game and the other is hitting .34 more homers per game, well, that means the difference (.46) are outs. And outs are bad. People tend to forget that part.

An out is worth negative-.27 runs.

So, you have to take away .12 runs away from the 2017 hitters because of those outs (.27 * .46 = .12).

And now, like magic: .48 runs (for the homers) – .12 runs (for the outs) = .36 runs.

Now look back up at 1993 — right, those singles were worth .37 runs. It’s almost exactly the same.

That’s how it evens out. Kind of cool, right?

Linear weights is fun to use. People have very strong feelings about them, good and bad, but they give you a good feel for the game. For example: Is it worth more to go two for six with two home runs or four for four with all singles?

You probably have an immediate answer that came to your head. Let’s do the math:

— 4 singles is worth (4 * .45) = 1.8 runs.

— 2 homers is worth 2.80 runs. Then you have to subtract the four outs (4 * -.27). So that means you have to subtract 1.08 runs.

That makes the two-homer day worth 1.72 runs or SLIGHTLY LESS than the four singles day.

You might disagree with that (and it’s so close that it’s almost no difference at all; plus it’s hard to entirely separate individual play from team play). But the thing linear weights does really well is give you a good feel for just how much an out costs a team. It’s so easy to forget: Not making outs is still the most important thing a hitter can do.

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13 Responses to Baseball’s balance in math

  1. Bryan says:

    Step 2: Batting Average of Balls in Play
    Single is worth 0.45, Out is worth -0.27 so if you hit a single 37.5% of the time and create one out the other 62.5% of the time your linear weight is 0.
    In 2017, 23 of 288 players have a BAbip of .375 or higher. In 2016-17 of the 299 players with at least 300 PA and have played in 2017 (excluding retired players like Ortiz) only Tyler Naquin .407, Tyler Flowers .393, Keon Broxton .379 and Alex Avila .375 are in the club.
    The 4 single example is largely meaningless when it takes peak Wade Boggs (Age 25-30, .371 BAbip, 1274 Hits) to get a null linear weight value from singles which is roughly a league average hitter. Ty Cobb and other players from 100+ years ago have higher career BAbip and/or long stretches, Boggs maintained one of the highest BAbip in the last century or so over a period of years that includes at least 1000 hits.
    Dee Gordon in 2015 had a .383 BAbip, 169 singles, 24 doubles, 8 triples, 4 HR, 25 walks, 2 HBP and hit into 6 double plays. His 116 OPS+ and 9.0 Rbat are complex calculations which peg him as above average (100 OPS+, 0 Rbat). Pujols in 2015 had a .217 BAbip, 85 singles, 22 doubles, 40 HR, 50 walks, 6 HBP and hit into 15 double plays for 118 OPS+ and 12.0 Rbat.
    Essentially they provided very similar value as batters. wOBA which is basically linear weights with values raised so an out is 0 has Dee Gordon at .337 and Pujols at .333 in 2015 tied for 57th and 69th respectively out of the 141 players who qualified for the batting title. At which point it’s that it’s really hard to teach someone to run really fast and get lucky with BAbip and take PEDs to double benefit the team (gain from improved performance, save money while you are suspended) as opposed to teaching someone to swing hard at everything, accept the strike outs and try to put a ball over the fence every week or so.
    Yes if someone could get 2 singles instead of a HR there is roughly a break even point either on the 4 singles vs 2 HR example in the article or the 84 more singles (+2 doubles, +8 triples) vs 36 more HR (+25 walks) of Gordon vs Pujols. A decent number of young hitters have proven able to mimic 2015 Pujols or better.
    Even Dee Gordon can’t match 2015 Dee Gordon, since the suspension he has .325 BAbip, 81 OPS+ and -14.0 Rbat. Pujols in 2016-17 has .260 BAbip, 108 OPS+ and 7.0 Rbat even though he’s 8 years older than Dee Gordon so much less likely to maintain his rate stats all else being equal. Pujols is one of the greatest players of all-time and Dee Gordon got a batting title because of some quasi-random decisions over a century ago on things like walks so all else isn’t equal but the really high BAbip going down and the really low one going up is not at all surprising.
    Information provided by Play Index.

  2. MikeN says:

    4 for 4, what happened to the other two plate appearances, which are now more likely since you haven’t made any outs?

  3. KHAZAD says:

    It is, of course, a tremendous advantage to swing to make hard contact early in the count. It is a smaller (but still significant) disadvantage to continue doing so with two strikes.

    Using Tango’s linear weights and breaking down the two seasons you mentioned to a level per 1000 PAs, there are 482 PAs where, in both seasons, the ball was put in play before two strikes. In those 482 PAs, the 2017 version of swinging hard early in the count is worth about 17 more runs. However, there were 104 PAs per thousand in 1993 where the ball was put in play before two strikes, but the count got to two strikes in 2017. Those 104 PAs were worth about 13 more runs. In the 414 remaining PAs, the 1993 version made up the other 4 runs with a more traditional two strike approach, putting the ball in play 27 more times.

    You can tell by the lack of decimals above that I did some rounding with the results, but I think if you run these by your friend you will find they are basically accurate.

    The point is it is a tremendous advantage to swing for harder contact, but only if you actually make that contact before you get to strike two.

    • Brent says:

      One thing that is forgotten in this is by swinging for hard contact early in counts, you create more walks for yourself. Obviously swinging for hard contact means you miss more often and the at-bat goes on, which may lead to a walk (or a strikeout). The batter who swings to make contact early in the count never gets to a walk count. One of the “skills” involved for players who are high walk guys is usually the “skill” of swinging and missing. (think about it, George Brett, Albert Pujols, Ichiro, those guys didn’t miss 2-0 pitches very often, they hit those pitches) Thus hitters like that never walk a ton because they rarely missed pitches in hitter’s counts. High walk guys (Adam Dunn, Jason Giambi) do.

      • KHAZAD says:

        This is anecdotal. In fact there is virtually no difference in walk rate between the two seasons, as it stands at about 8.65% in both 1993 and 2017, with both seasons having slightly less walks (about 84 per 1000 PAs each) in PAs that reach two strikes.

  4. Dan says:

    Re this and the previous article: Good work, but you get an important number wrong: 2017 OBP is 10 points lower than 1993, not (as you say) 17. The difference is fully accounted for by batting average. The walk rates in the two seasons (8.7%) are the same.

  5. IRAM01 says:

    Joe, I think you have an error in the calculation for 2 for 4 with two homeruns. You only have two outs: 2 homers is worth 2.80 runs. Then you have to subtract the TWO outs (2 * -.27). So that means you have to subtract 0.54 runs. This leads to 2.26 runs, i.e., about 25% more runs.

  6. Mark Daniel says:

    How is a 2 homer day worth 1.72 runs, since at minimum 2 homers results in 2 runs? I don’t care how many outs a player makes, 2 runs will score if you hit 2 home runs.

    • invitro says:

      I think the 1.72 means 1.72 runs above average. It makes sense, and there is this: “This gives you the runs above average produced by each of these kinds of events, also known as linear weights”, followed by “For wOBA, we have the runs above average for walks (0.29), HBP (0.31), singles (0.44), doubles (0.74), triples (1.01), and home runs (1.39)” which are almost the numbers Joe uses. This is from .

  7. Pat says:

    Off-topic, but if anyone reading has Bill James 2000 Historical Abstract handy, go read the entry for Cy Young and see Bill get into the math of baseball, especially as it relates to the Golden Ratio. I love that part.

  8. Matt Trowbridge says:

    To me 4-for-4 with four singles vs. 2-for-6 with 2 HRs comes down to the rest of your team. The worse your team is, the more you need home runs, even if it comes with a Dave Kingman batting average. But if you play in a lineup loaded with great hitters, a Wade Boggs type is far more valuable. … When Sammy Sosa was MVP in 1998, he had 292 runs plus RBI vs. 277 for Mark McGwire. But McGwire reached base 48 more times and made 89 FEWER outs. … Their true value wasn’t even close, but the writers picked the wrong MVP.

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