Exercise 1: Why Negative Variance Proves a Bug, Not Just Inaccuracy — Possible Solution
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THE MATHEMATICAL FACT THAT MAKES THIS DIAGNOSTIC
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Variance is defined as an average of squared deviations from the
mean: Var(X) = average of (x - mean)^2 across all data points. Every
single term being averaged is a square, and a square of a real
number can never be negative. An average of non-negative numbers can
never itself be negative. This isn't a property of any particular
dataset - it's true for every possible dataset, by the definition of
variance itself, before any computation is even attempted.

WHY THIS DISTINGUISHES "BUGGY" FROM "SOMEWHAT INACCURATE"
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An approximate variance calculation that returns, say, 0.06 when the
true value is 0.0673 is still inside the space of mathematically
valid variance values - it's simply not precise. That's ordinary,
expected floating-point imprecision, the kind Chapters 1-4 already
established happens in nearly every computation. A NEGATIVE result,
by contrast, isn't merely imprecise - it's a value that no correctly
functioning variance computation, run on any real dataset, could
ever mathematically produce. Getting a result outside the entire
range of valid outputs is qualitatively different from getting an
approximately-correct value inside that range.

WHAT THIS TELLS US ABOUT THE NAIVE FORMULA SPECIFICALLY
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Because negative variance is mathematically impossible in principle,
observing it directly proves that the computation's own rounding
error became larger than the true quantity being computed - the
accumulated error in E[X^2] - (E[X])^2 exceeded the size of the true
variance itself, and swamped it entirely. This chapter's own verified
example showed exactly this: naive_var = -0.0625 against a true
value of about 0.0673 - the rounding error was actually larger in
magnitude than the correct answer.

WHY THIS WORKS AS AN ANSWER
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The explanation grounds the argument in the actual mathematical
definition of variance (an average of squares, which forces
non-negativity), distinguishes "outside the valid range entirely"
from "inside the valid range but imprecise," and connects the
conclusion back to this chapter's own verified -0.0625 result as
concrete evidence rather than a hypothetical.