Thanks for responding. I’m still trying to wrap my head around the equation as well as this site. I published a Google spreadsheet with findings for over 100 seasons:
On the spreadsheet I include the actual denominator in the (SqRt OBP*1000)*(SqRt SLG*1000)/x = SqRt Run Env., as well as the results and % accuracy when I used two fixed numbers (13.54 and 13.49). Because of my “disadvantages” with math the chart deals with OBP and SLG already multiplied by 1000. I also included your formula with a twist - 4.5 seemed to run high, but when I used 3.6 things got a bit more accurate. Again, the old trial and error method which I could never be broken of.
(square root(OBP*1000))*(square root(SLG*1000))/13.5=square root of the Run Environment.
Square both sides:
OBP*1000*SLG*1000/182.25 = run environment
OBP*SLG*5487 = run environment
There’s some kind of mistake here since the 5487 is way too high. But basic runs created is OBP*SLG*AB and the average number of AB in a game is something like 68. So a formula like:
OBP*SLG*68 = run environment
I’m not sure why 5487 is too high a number as it represents a constant that kept coming up in the formula. It took me awhile to remember some Algebra - I appreciate you leaving your steps.
However! The average number of AB in a game will be closely related to the OBP. Assume that AB=PA*0.9. Then assume there are 50 OBP-outs per game. (Why 50? A double play counts as a single “OBP-out” but 2 actual outs. A caught stealing, etc. is zero OBP-outs and 1 actual out. An error is 1 OBP-out and 0 actual outs. There are 54 actual outs per full 9-inning game, but there are games without a bottom of the 9th, extra-inning games, rain-shortened games, and so forth. Put it all in a blender and you get 51 OBP-outs per game for the AL so far this year. 50 is close enough.)
If X is the number of PA per game, the number of OBP-outs per game will be (1-OBP)*X. So (1-OBP)*X=50 and X=50/(1-OBP). Thus AB=45/(1-OBP).
The formula I would use is this:
[OBP/(1-OBP)]*SLG*45 = run environment
The constant 45 probably needs to be tweaked somewhat.
I’m following your logic for the most part - and it’s nothing on your end, just me entering a new field of thought. I’m guessing the number has to be adjusted for the deadball era on an almost different scale because of fielding and stolen bases, which may explain why the early years come up short. 1994-95 are the largest percent differences on both systems.
I’ll continue analyzing your method and see where the difference lies between 4.5 and 3.5 and see how that number will specifically relate to AB and all the factors we discussed. Of course, this makes my task of neutralizing John Beckwith and George Scales all the more difficult as league totals are lacking for most of this. I had to fudge league OBP for my estimates, though I did it somewhat fairly. At least, scientifically enough to run a sim.