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The proof is by definition.

That is the definition of median. You might be confusing the definition of median with mean or mode.

Thank you I was

That's what the median is! The middle number in a sorted list.

It’s the definition of median. It doesn’t need a proof.

If you want to define the median in a way that generalizes to other situations, this is actually not the way to go! Here's a more general definition: a median of the numbers {x_1, ... x_n} is a number m that minimizes the sum of the absolute deviations, |x_1 - m| + |x_2 - m| + ... + |x_n - m| . Medians are generally not unique via this definition, but they can then be defined for things like sets of points in the plane, in space, etc. (cf. geometric median); the traditional definition is the average of all of the medians under this more general definition.

sorted

It's been 20+ years but this still sounds familiar. Might just be the "that's sounds right so I'll assume I knew it" bias that makes educated people susceptible to subtle propaganda, though. :)