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The intuition should be that "Half the population has below average X" is untrue for the vast majority of situations, because most real things have a skewed distribution.

AFAIK IQ is a peculiar case because of how IQ is defined.

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A lot of natural processes produce results that are normally distributed. So given a large sample, a continuous value range, no bounds (or far away bounds) and a lack of obvious skew the intuition "half the population is below average" has a good chance to be (approximately) true.

Of course IQ is a made up thing that we just defined to behave that way.

Yes, the obvious counter-example is that almost the entire population has more than the average number of legs.

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