by Kevin Harlow
July 21, 2003
I derived the following equation in OPS Junk, Part I:
(RC/PA) = 0.3*OPS - 0.1
I think there is some utility in this simple equation for the back-of-the-envelope calculator. In particular, what follows is a quick-and-dirty method to roughly convert a player's OPS to Batting Wins Above Replacement or Batting Wins Above Average. First the equations and and an example and then I'll show why it works.
BWAA = [(OPS - .750)/.1] * [PA/100] / 3
Example:
Player A: 850 OPS, 700 PA
Player B: 950 OPS, 500 PA
Who is more valuable (in terms of BWAR)?
BWAR_A = [(.850 - .650)/.1] * [700/100] / 3
BWAR_A = (2) * (7) / 3 = 14/3 = 4.7
BWAR_B = [(.950 - .650)/.1] * [500/100] / 3
BWAR_B = (3) * (5) / 3 = 15/3 = 5
Answer(s): 1) Player B by 0.3 wins; 2) They're too dang close for this back-of-the-envelope calculation to show a real difference.
The calculation is pretty simple to perform in your head if you think i) hundreds of OPS above reference; ii) hundreds of PA; iii) i * ii /3
RC = (0.3*OPS - 0.1) * PA
RC_Replacement=RC_R = (0.3*OPS_R - 0.1) * PA
Runs Above Replacement = RAR = RC - RC_R = PA * 0.3*(OPS - OPS_R)
Batting Wins Above Replacement = BWAR = RAR / (Runs/Win)
BWAR = [PA * 0.3*(OPS - OPS_R)] / (Runs/Win)
The form of the above equations is the same regardless of the reference performance level. When the reference performance level is league average, the following equation results.
BWAA = [PA * 0.3*(OPS - OPS_A)] / (Runs/Win)
So what you need to have is a rough estimate of the average and replacement level OPS, and (Runs/win). An approximation for (runs/win) is two times the league average runs per game (LgRPG). LgRPG has historically been ~4.5 RPG, making the (Runs/win) = 9. The league average OPS is about .750. The replacement level OPS was chosen to be .650 since that is a nice round number and it makes the pythagorean (with exponent of 2) replacement winning percentage between 35-40% (37.7%).
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