No, the word average is a place holder for measures of central tendencies (like mean, median and mode).

Any symmetrical distribution will have the mean equal the median.

Not exactly. This pathological distribution has no mean. https://en.wikipedia.org/wiki/Cauchy_distribution

You got me—funny thing is I took a couple analysis courses in school so I should have been more careful making such a strong statement.

But that’s the definition of a standard distribution? Intelligence is not, at least I think.

AFAIK "standard distribution" is not a defined mathematical term; usually it's shorthand for standard normal distribution.

Can't comment on intelligence distributions.

"standard normal distribution" is a normal distribution with mean 0 and s.d. 1.

To talk about whether intelligence follows a (one-dimensional) normal distribution we have to assign a number to it. That number is usually IQ, but by design the raw score is transformed to make IQ scores follow a normal distribution.

So it is trivially true.

If we want to go beyond that, what does it even mean to say, for example, "twice as smart"?