by Kevin Harlow
May 26, 2003
Just a little quick algebra to show the one-to-one relationship between ERA+ and a support neutral win/loss percentage (SNWL%). Note that SNWL% is not from Baseball Prospectus but just uses end-of-season pitcher ERA and league ERA and park factors.
First, we have ERA+:
ERA+ = 100*(PF/100)*LgERA/ERA
ERA+ = PF*LgERA/ERA
Next, we have SNWL% (assuming "Pythagorean Theorem" run-to-win converter):
SNWL% = [(PF*LgRA/100)^2] / [(PF*LgRA/100)^2 + RA^2]
If you make some assumptions about equality of defenses and non-ER park factors, the above is
SNWL% = [(PF*LgERA/100)^2] / [(PF*LgERA/100)^2 + ERA^2]
SNWL% = [(PF*LgERA/100/ERA)^2] / [(PF*LgERA/100/ERA)^2 + 1]
That leads to the following two formulas:
SNWL% = [(ERA+/100)^2] / [(ERA+/100)^2 + 1]
ERA+ = 100 * sqrt[SNWL%/(1-SNWL%)]
An important thing to notice is that SNWL% is only a function of ERA+. So regardless of whether or not it is easier to obtain a high ERA+ in a high-offense era, from a value perspective a higher ERA+ leads to a higher SNWL% and a high ERA+ produces the same SNWL% in both high-offense and low-offense eras.
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