TL;DR of OPS Being Bad
Such was the question posed by a Twitter user in a Tom Tango thread yesterday (well, the question was "What's the…”). My response was:
(TB*PA + TOB*AB)/(AB*PA)
This, as you may recognize, is the formula for OPS—what actually happens mathematically when you add two fractions with different denominators (for this post, OBA = TOB/PA and SLG = TB/AB; later, W will encompass walks and hit batters).
When viewed in this manner, it becomes clear that OPS works (such as it does) by accident. No one in their right mind would ever multiply total bases by plate appearances and add times on base times at bats.
This is all ground I've covered before, but with a little calculus you can rewrite OPS as an equation of the form:
OPS = (x1*S + x2*D + x3*T + x4*HR + x5*W)/PA
This seems like a perfectly reasonable form for a measure of offensive productivity to take, but only if the coefficients are set in a reasonable manner. Unfortunately, due to the way OPS mixes denominators, the coefficients are defined solely by the variable A, which is the proportion of plate appearances which are at bats (A = AB/PA).
x1 = 1 + 1/A
x2 = 1 + 2/A
x3 = 1 + 3/A
x4 = 1 + 4/A
x5 = 1
There might be good reason to want an offensive metric to assign different values to the same event for two teams or players being evaluated (e.g. a dynamic run estimator for the former, a theoretical team model for the latter). There is no reason to suspect that doing so on the basis of the proportion of plate appearances that are at bats is one of them.
I'm not exactly a fan of OPS+, but if you interpret a 118 OPS+ as "18% more productive than a league average hitter" rather than "18% higher OPS than a league average hitter", it is a correct statement (at least to the extent that OPS+ is a predictor of team runs scored). The negative values are an unavoidable consequence of a linear run estimator applied at extremes of offensive ineptitude, although I certainly understand why one might find them obnoxious.
I wrote a little bit about the interpretation of OPS+ in response to Bill James' comments in his 2023 Handbook (RIP) last year: https://walksaber.substack.com/p/rehashing-runs-created-and-ops
While OPS is bad enough, I cringe most at the definition of OPS+ which is OBP/(leave average OBP) plus SLG/(league average SLG) minus one. How did that originate? People erroneously talk about percentages (e.g., an OPS+of 118 they say is 18% above average). This obviously breaks down for very low OPS+ which can theoretically bottom out at -100. There are pitchers with a negative career OPS+ like Sandy Koufax with a -39 OPS+. What percentage worse is he than average? 139% worse?