In a normal distribution, like lets say a large population, the mean, median, and mode are all equal. I don't see a problem with the quote.

I always point that out too. And someone always points out that with distributions average and median are the same. That still doesn't make it right. What's worse is it's a quote about intelligence, so you would think it would use the correct word. Maybe that's the joke, it's summer dumbed down and smug in the knowledge that everyone that hears it thinks "thank god I'm above average!"

You can absolutely use the word average to refer to the median when you are speaking in a non-academic context.

https://www.merriam-webster.com/dictionary/average https://en.wikipedia.org/wiki/Average

It is on you to prove that human intelligence does not follow a normal distribution, if you find that funny and incorrect.

Median is one type of average.