How would this House Rule for combined Advantage/Disadvantage Impact Rolls?

Question

According to the rules for Advantage/Disadvantage:

If circumstances cause a roll to have both advantage and disadvantage, you are considered to have neither of them, and you roll one d20. This is true even if multiple circumstances impose disadvantage and only one grants advantage or vice versa. In such a situation, you have neither advantage nor disadvantage.

(PBR 60)

While I understand the main point of this is to simplify things and avoid the insane stacking of individual bonuses found in 3.5e, it's disappointing that stacking advantage and disadvantage is the same as having neither. I'm thinking of a house rule like this:

If circumstances cause a roll to have both advantage and disadvantage, instead roll three d20 and take the median value.

My gut says that this would keep the average roll at 10.5, but tend to cluster rolls around 10 instead of the even distribution of a single roll, but I'm not sure how to actually calculate the distribution. Regardless, rolling dice is fun, so this seems like a way to keep things balanced while still keeping the thrill of rolling extra dice that comes from advantage or disadvantage.

Would this method actually keep the average output the same as 1d20? Are there any edge cases or unforeseen circumstances where this would be more advantageous or disadvantageous than it should be?

Answer 1

This makes your roll have the same average, but makes the variance much smaller

Here's an anydice graph. You can see the change makes your distribution a bell curve rather than uniform. While the odds of getting at least a 10 or 11 and the odds of getting at most a 10 or 11 are basically the same, the odds of rolling a 5 or less and the odds of rolling a 15 or more are 10 percentage points (~30%) lower.

Essentially, this is a less-static version of having people take 10 on things. They are unlikely to be able to do anything they couldn't do reliably but unlikely to fail to do anything they can usually do reliably.

This makes Super Luck even more Super

One of the Lucky feat's primary strengths is that it lets you turn any roll with disadvantage into super advantage: instead of rolling two dice and taking the worst, you roll three dice and take your choice. With this rule you now have a better option-- turning super mediocrity into superduper advantage: instead of rolling three dice and taking the middle, you would roll four dice and take your choice. Considering advantage and disadvantage are usually pretty easy to give yourself, especially in combination with each other, expect everyone to take the Lucky feat and important rolls to look like this: