Exercise 3: Sample vs. Population Variance in Practice — Possible Solution
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WHY n-1 IS ALMOST ALWAYS THE RIGHT CHOICE HERE
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The data engineer has 10,000 user sessions out of millions that
actually occurred - this is a SAMPLE of the full population of
sessions, not the complete population itself. Per this chapter's own
Bessel's-correction explanation, using the sample's own mean (computed
from just these 10,000 sessions) as a stand-in for the true population
mean systematically underestimates the true variance if divided by n.
Dividing by n-1 instead corrects for that bias, producing a more
accurate estimate of the TRUE population variance - the variance
across all the millions of sessions the engineer doesn't actually have
direct access to. Since the engineer's real goal is almost certainly
to draw conclusions about user behavior in general (the full
population), not just describe these exact 10,000 rows in isolation,
n-1 is the correct choice.

THE ONE CIRCUMSTANCE WHERE DIVIDING BY PLAIN n IS CORRECT
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Per this chapter's own explanation, dividing by n is the
mathematically correct choice specifically when the dataset in hand
IS the entire population, not a sample drawn from a larger one - i.e.,
if those 10,000 rows were literally every session that had ever
occurred (nothing left out, nothing yet to happen), there would be no
larger population being estimated, and no bias to correct for. In that
specific case, dividing by n gives the exact, true variance of that
complete dataset directly.

WHY THIS WORKS AS AN ANSWER
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The explanation identifies the 10,000-out-of-millions scenario as a
genuine sample (not a full population) and applies this chapter's own
Bessel's-correction reasoning to justify n-1 as the right choice, then
separately names the one specific condition (the data being the
complete population, not a sample) under which n itself would actually
be the mathematically correct divisor instead.