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.
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?