Math: Incremental Improvement from Raising a Skill by One Level

PDF in blue, using the axis on the left
CDF in orange, using the axis on the right.

I was thinking about something last night about the way we generally present the bell curve for a roll of 3d6 as a way to distinguish it from the flatline of a single d20. We then take that probability density function and integrate to get the cumulative density function, or the graph that shows our success of rolling "x or under." These visualizations, I think, are useful, and they aren't wrong, but I don't think the information it conveys to a player in helping making strategic decisions about spending points is showing the whole picture, so I made a new graph that shows different information.

The information this graph shows is how much of an improvement each step makes. You might think the cumulative density function already shows us that, and it does, but from a different, less personal perspective. It shows the cumulative chance of success versus failure, but it doesn't focus on the increase, or if we are speaking to math terms, the slope. This increase in probability of success taken in a relative light instead of an absolute light shows us how much it helps to improve a skill level.

The amount of improvement per
increment of a skill level.

Take for example, the first graph again. The odds of rolling a 3 and the odds of rolling either a 3 or 4 are both pretty small numbers. about 0.46%, and 1.85% respectively. However, if we look at this in a different light, that is a 300% increase in probability of success. The odds of succeeding have quadrupled.

Technically, those numbers are treated a bit funny by GURPS because they are in the weird tail-end that has special case rules for critical success, so let's take a look more in the middle of the range that more matches something we are liable to see.
The odds of rolling exactly an 8 are around 9.72%. The odds of rolling an 8 or less is around 25.93%. An 8 is a pretty nominal starting point for a nominal character's dodge stat.
(10HT + 10DX)/4 + 3 = 8 Dodge
Now, what are the benefits of improving that to 9? The odds of rolling exactly a 9 are around 11.57%, and the odds of rolling 9 or under is 37.5% exactly. By the way:
37.5% = 25.93% + 11.57%
So, in one light, this looks like something around a 12% improvement, but the nuanced truth is that we have just decreased the odds of failing by 12% and increased the odds of succeeding by 12%.
In actuality, improving our odds of 25.93% by an additional 11.57% is a 44.64% improvement.
11.57%/25.93% = 44.64%
That's a tremendous return on investment and one that is kind of lost in the regular charts. A person with 9 dodge is 45% safer than someone with 8 dodge. The chance of someone with 9 dodge avoiding harm is 37.5%, but that is 45% better than the odds of dodging with a dodge of 8, which is 26%.

Other Thoughts

For giggles, I compared the improvement function on the 3d6 CDF to that of the improvement function on a 1d20 CDF. Interestingly, the function is almost literally f(x) = x-1. It has a R2 value of 1 in fact. I feel like this has to mean something about the relationships of numbers when rolling a single die, but I don't quite grasp it. Any mathematicians know what is happening there? Well, for more of my thoughts on these distributions, I made this post a long while back, check that one out too.
This post totally ignores the impact of positive and negative situational modifiers, but extrapolating a bit, the size of their impacts are a lot bigger than they might first appear as well. I'll leave that as an exercise for my dear gentle readers.