Thank you for expressing very clearly what I wasn't able to sort out in my mind. I guess the question for Sean is, "Which career value is more realistic?" I suspect many of us will be using the weighted average to assess segments of careers (like we used to use the harmonic mean), so I'd argue that the weighted average should also be used for the career value -- and it moves Walter Johnson JUST ahead of Lefty Grove on the career list, which makes for a wonderful debate.

One example in favor of the weighted-average career calculation:

Year W-L IP ERA* ER LgER R/9

1 20-20 360.0 100 80 80 4.00

2 30-10 360.0 150 40 80 4.00

TOT 120 160 125 ERA*

Change Year 2's environment:

Year W-L IP ERA* ER LgER R/9

1 20-20 360.0 100 80 80 4.00

2 30-10 360.0 150 80 160 6.00

TOT 160 240 133 ERA*

Weighted average for both scenarios is 125, which more consistently encapsulates "360 innings of average, 360 innings of 50% better, therefore career total is 25% better."

It'll be interesting to see which way B-R decides to go.

]]>The weighted average of each season equaling the career value relies on the league ERA being the same in each season that you are averaging into the total. ERA+ is ERA/lgERA (and then flipped to work down from 2 instead of up from 0, but it works out the same as if you treat it as working up from 0). The career value is total ER divided by total league ER, which is the same as taking the sum of the numerators (pitcher ERA) weighted by IP, divided by the sum of the denominators (league ERA), also weighted by IP. As long as the denominator (i.e. the league ERA) in each fraction (i.e. each individual season ERA+) is the same, then the weighted average of each fraction simplifies to the weighted sum of the numerators divided by the weighted sum of the denominators, which is career ERA+. If the denominators (league ERA) are different, then the weighted average no longer simplifies to the weighted sum of the numerators divided by the weighted sum of the denominators, so the weighted average of each season ERA+ is not mathematically equivalent to career ERA+ if the league ERA is not the same in each season.

However, it works out to be fairly close in practice (such as 132 vs. 133 for Walter Johnson). Taking the weighted average is not actually calculating the total ERA+, but it is a good estimate. Also, the same problem existed with averaging multiple ERA+s before because you still had the same issue of the league ERAs being different in different years/ballparks. It's just that now you don't have to take the harmonic mean to get the same issue.

]]>there should be no "league" before the 2 in that sentence.

]]>OPS+ relates to runs in the same way as the new ERA+. Both take the form of:

(lgAVG + RAA) / lgAVG

where lgAVG is the average number of runs (or runs allowed) over the player's PA/IP. OPS+ uses the form it does instead of (OBP/lgOBP + SLG/lgSLG)*50 or OPS/lgOPS because it is designed to model percentage changes in runs, not OPS. If it were instead designed so that going from a 100 OPS+ to 110 represented a 10% increase in OPS, it would no longer represent a 10% increase in runs, because a 10% change in OPS does not translate to a 10% change in runs. OPS+ had to choose to scale to one or the other, and it chose runs since runs are more useful. So the construction of OPS+ gives it the same relationship to runs as the new ERA+ has (and also more or less the same relationship to wins). A 110 OPS+ team will tend to score 10% more runs than average, and, given that it allows an average number of runs, win 10% more games than average. It's the same as with ERA+.

]]>OPS+ is already linear in OBP and SLG (as opposed to linear in the inverses of those things like the old ERA+ was). However, a certain % increase in OPS+ does not give the same % increase in wins -- an OPS+ of 110 doesn't mean a team wins 10% more, it's more like 5% more).

If you made a new OPS+ as (OBP/lgOBP + SLG/lgSLG)*50, then % changes in OPS+ are the same as % changes in OPS (assuming OBP and SLG scale the same), and dividing OPS+ by 2 would give an approximate WPct, just like the new ERA+. I'm not sure why this wasn't done originally, I think (somebody will correct me if this is not true) the reason was to expand the scale of OPS+, i.e., allow for more granularity in OPS+. But if you want to make a new OPS+ with a similar relationship to wins as the new ERA+, that would be the logical way to do it.

For either the new ERA+ or a potential new OPS+, the relationship to WPct breaks down at extreme values, since it's a linearized approximation.

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