No flips for you!
Sigh. The screenshot you presented should have clued you into a problem with what you did. The scaked and recentered 3d6 curve has values from -5 to 25 while the d20 is 1 to 20. All you've done is stretch a normal cumulative dustribution and note that, if you strech it enough, the middle part looks straight(ish). You're ignoring ~1/3 of the data points to do this.Have you looked at the graphs I've been linking to, or not?
How math works is you do steps and results come out.
Do I have to take a screen shot? I have to take a screen shot. Fuck.
View attachment 117346
So here we have the CDF (cumulative distribution) of 1d20 and the CDF (cumulative distribution) of 3d6 with different averages and standard deviations normalized.
The 1d20 curve is a line. The 3d6 curve is the set of black points. Notice how the 3d6 curve is close to, but not exactly on, the 1d20 line. It only differs significantly at the 5% "critical hit/miss" cases that correspond to 1 and 20 on the d20 roll.
I horizontally scaled 3d6 by a factor of 2, which corresponds to "bonuses and penalties are twice as large, conceptually, in a 3d6 based situation".
So yes, that is how that works. The distributions are similar in CDF, because you can see it. Yes, one is a flat distribution and the other is a normal(ish) one, but we aren't playing "can you roll a 7", we are playing "can you roll a 7+" when we play D&D. And "can you roll a 7+" corresponds to the CDF (the integral) of the distribution.
And when you integrate things, the differences between a flat distribution and a curved one fade away pretty fast.
This isn't "mathturbation", because I actually checked my results. I even shared links to those results being checked. I am not sure why I expected people to actually click on those results before saying "this is bullshit".
Anyhow, here is the results inline.
Quite possibly a slightly different value than "2" would be more correct once we neglect tails -- a different value than "2" would correspond to a change in the slope of the 3d6 part of the graph, and making it slightly less steep might improve the match (except for the tails). But 2 is so close I really don't care.
This is like blowing up a circle to a large enough circumference that a close look at a tiny part of the arc looks like a straight line. But, despite doing this, a circle is not a straight line. This is why when you do math, you really need to understand what your doing -- what assumptions are necessary. Just doing math doesn't mean you'll get the right answer. Especially with stats.