Since intelligence – and I don't mean IQ but intelligence as a trait just like height or even your reaction speed or whatever – is more than likely normally distributed just like the vast majority of other traits, the average is the median.

If we're going to be pedantic, you mean that it's possible for the median to equal the arithmetic mean. The "average" is a number that represents a set of data, and could be the median, arithmetic mean, geometric mean, and several other values.