Exercise 1: Why an Outlier Spikes RMSE Much More Than MAE — Possible Solution
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WHAT THE CHAPTER'S OWN WARN-BOX CLAIMS
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Per this chapter's own warn-box, "if a genuine outlier like ds1-9's own
vintage Jaguar ended up in the test set, RMSE would spike sharply while
MAE moved only modestly — a direct, concrete consequence of squaring:
one enormous miss contributes its squared value to MSE (and therefore
RMSE), but only its own plain, unsquared size to MAE."

HOW MAE TREATS A LARGE ERROR
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Per this chapter, MAE is "average |predicted − actual|... Every error
contributes to the average in exact proportion to its own size." If the
Jaguar's real price is, say, $85,000 and the model badly mispredicts it
by $40,000, that single error contributes exactly $40,000 to the sum
being averaged for MAE — large, but proportional, exactly as large as
the error itself and no larger.

HOW RMSE (VIA MSE) TREATS THE SAME LARGE ERROR
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Per this chapter, MSE is "average squared error," and RMSE is "simply
MSE's own square root." That same $40,000 error doesn't contribute
$40,000 to the sum behind MSE — it contributes $40,000², i.e.
1,600,000,000, an enormous number compared to what a handful of
ordinary, few-hundred-dollar errors would contribute individually.
Squaring a large error doesn't just make it bigger — it makes it
disproportionately, quadratically bigger relative to the small,
ordinary errors elsewhere in the same dataset.

WHY THIS PRODUCES A SHARP RMSE SPIKE BUT ONLY A MODEST MAE MOVE
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Because the one enormous squared error can dominate the entire sum MSE
is averaged from, MSE (and therefore RMSE, its square root) responds
very sharply to even a single such outlier — the one Jaguar-sized error
can outweigh the combined contribution of many smaller, ordinary
errors. MAE, by contrast, only ever adds that error's own plain,
unsquared size to the total — a large number, but not one that swamps
every other error's own contribution the way its squared version does
for MSE/RMSE. This is precisely why a large gap opening up between MAE
and RMSE is a meaningful signal, not noise: it specifically indicates
that a small number of unusually large errors — not a broad, even
spread of moderate ones — are driving the overall error total.

WHY THIS WORKS AS AN ANSWER
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It traces the Jaguar example through both formulas explicitly — showing
the identical $40,000 error contributing its own plain size to MAE's
sum but its squared, vastly larger value to MSE/RMSE's sum — and
explains why that structural difference is exactly what makes outliers
spike RMSE sharply while barely moving MAE.