Conversely, a 14 is as far from average as a 6 is. Does a 14 mean that you are as much more intelligent than a commoner, than a commoner is compared to an ape? Because that would be super-genius level.
@Elfcrusher et al. (in case no one pointed it out...)
From 1E MM1, MM2, Field Folio, and Deities & Demigods (might even be in other books...):
Unfortunately, this doesn't
directly translate to the normal distribution of the 3d6 bell-curve so you can't simply compare it to the standard curve for I.Q.
Anyway, notice (most importantly IMO) is states "In monsters" first thing. You can use the same table for PCs, but with their higher average (10.5 with 3d6, higher with other methods) it isn't quite the same IMO.
Now, I know 5E doesn't make the following assumption, but I like it and treat it as such when it comes to ability scores in 5E.
Each +1 modifier does translate directly into a one standard deviation increase in ability or increase in relative ability.
Ex. INT 14 (with +2) would be the equal of I.Q. 130 (2 SD above the mean), which is pretty smart and frankly research has shown with this I.Q. you can learn/understand/handle just about any task.
EDIT: with this concept, the actual bonus should also be cumulative, i.e. +1 => +1, +2 => +3, +3 => +6, +4 => +10, and +5 => +15. This way in a contested check for someone with -2 vs. +5 the modifiers would be -3 vs. +15, so the +5 would be nearly guaranteed to win.
----------------------------------------------------------
Anyway, to the OP:
You could remove ability scores in 5E and just allow PCs to select one thing to be great at, one good at, and the rest average or whatever.
For example, a fighter would be great at combat, good at saves, and average with skills, magic, or whatever else.
A rogue would be great at skills, good at combat, and average will saves, magic, etc.
That is just a quick-and-dirty idea, but I think you get it.
What a PC is great at uses double proficiency bonus, good uses 1.5, and average just gets proficiency bonus.
Something like that.
shrug