I think a lot about expected value. Expected value is just a fancy word for the mean (which is a fancy word for the average). I think very little about the median or mode. I think it’s kind of interesting to think about why.
I work in the stock market. If I told you that there’s a 25% chance that apple stock was going to be worth \$200 and a 75% chance it would be worth \$160, what is the most you would pay for 1 share? I claim the “right” answer is $0.25 \cdot \$200 + 0.75 \cdot \$160 = \$170$. In other words, I think the “fair” value of the stock is $170. That also happens to be the mean. Why didn’t I pick the median or the mode?
I think it comes down to how bets work in the stock market. In the stock market, if you pay \$X for something and it turns out to be worth \$Y you get (or pay) \$(Y - X). When bets work that way, the optimal bet to make is to buy below the mean and to sell above the mean. The mean is the value where you don’t expect to make or lose money regardless of whether you buy or sell. It’s “fair”.
What if bets worked differently? Right, that’s where I was going with this.
What if the stock market worked totally differently? What if you had to make a guess at where the stock was going to end up at the end of the year. For simplicity, let’s say stock prices always got rounded to the nearest dollar. If you guesed right, you get \$100. If you guessed wrong, you get \$0. Now, assuming aaple stock had the same possible outcomes (25% chance of \$200 and 75% chance of \$160), what would you bet?
\$160 of course. You’d bet the mode! All that matters is whether you’re right or not, and the mode is the most likely value to be right.
Ok, but that seems really weird and contrived. I kind of agree, but isn’t that kind of how horse races work? You just bet on a horse and you get money if you’re right? Ignoring all the complexity of odds, you’d want to just bet on the mode (the horse that’s most likely to win).
Now let’s come up with a betting structure for which the median is the “fair” value. I even bet you’ve used this one before with your friends. The way the bet works is that you each put up \$20 and guess the value of something. Whoever is closer gets the money.
In this case, you should bet the median! Half the values are lower than you, half the values are higher than you. Whatever your opponent guesses will be right less often than your guess. It’s optimal!
I don’t have any insightful grand finally here. I just found it interesting that I almost never think about values that aren’t “the mean” (although I do think a lot about variance and correlations) and I think this is a pretty plausible explanation as to why. If I was in a business where “closest wins”, I bet I would care a whole lot more about the median than the mean.