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> Half of the population is below average intelligence

By definition :)

Only if the distribution is symmetric. [0]

Extra pedantic mode: if the population is an odd number, the number of below average intelligence will not equal the number of above average intelligence.

[0] https://en.wikipedia.org/wiki/Symmetric_probability_distribu...

>if the population is an odd number, the number of below average intelligence will not equal the number of above average intelligence

I'll leave counting as an exercise for the reader :D

Not if there's exactly one person of exactly average intelligence. :-)

Or any odd number :)

You're right that if the set is of {99,100,101} then of course 100 falls into neither category and each category has cardinality 1, but we're talking about real-valued variables, so I don't think that's a worry. I imagine it happens 'almost never' (except where n=1).

[0] https://en.wikipedia.org/wiki/Almost_never

Good point :-P

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