Exercise 2: Why R² Is a Genuinely Different Kind of Question — Possible Solution ==================================================================== WHAT MAE/MSE/RMSE ALL MEASURE, PER THIS CHAPTER ------------------------------ Per this chapter, MAE is "average |predicted − actual|," MSE is "average squared error," and RMSE is "√MSE." All three share the same underlying shape: each one takes the SIZE of the model's own individual prediction errors and averages them (in one form or another) into a single number. Each is fundamentally asking "on average, how far off is this model's own prediction from the truth?" — just with different weighting of large vs. small errors. WHAT R² MEASURES INSTEAD, PER THIS CHAPTER ------------------------------ Per this chapter, "R² doesn't measure error size at all — it measures what proportion of price's own total variance (ds1-6's own vocabulary, directly) the model actually accounts for." Rather than asking "how far off, on average, are the predictions," R² asks "how much of the real, underlying spread in price does the model's own predictions actually capture, compared to the total spread that exists in price across the whole dataset?" This is a proportion (a fraction of variance explained) rather than a size-of-error quantity in the target's own units. WHY THIS IS A GENUINELY DIFFERENT AXIS, NOT JUST A DIFFERENT FORMULA ------------------------------ MAE, MSE, and RMSE are all interconvertible in spirit — they're variations on "average error size," differing only in how they weight large vs. small errors, and all three are expressed either in the target's own units or a squared version of them. R², by contrast, isn't measured in dollars or dollars-squared at all — per this chapter, it ranges from a theoretical maximum of 1.0 (all variance explained) down through 0 (no better than always predicting the mean) and even below into negative territory (worse than that trivial baseline). This dimensionless, proportion-based nature is precisely what makes R² not simply "a fourth way to measure error size" but a fundamentally different kind of measurement altogether — one comparing the model against a baseline (the mean) rather than measuring raw prediction distance. WHY THIS DISTINCTION MATTERS PRACTICALLY ------------------------------ Because R² is scale-free (a proportion, not a dollar amount), it can be compared meaningfully across completely different problems with completely different units — an R² of 0.82 predicting car prices in dollars and an R² of 0.82 predicting, say, a house's square footage mean roughly the same thing (82% of variance explained) despite the two target variables having nothing in common numerically. MAE/MSE/RMSE values, being tied to the target's own original units, can't be compared across problems with different units at all. WHY THIS WORKS AS AN ANSWER ------------------------------ It groups MAE/MSE/RMSE together as variations on the same underlying "average error size" question, using the chapter's own definitions, and contrasts that against R²'s own definition as a variance-explained proportion — a genuinely different axis of measurement, not merely a fourth variant of the same idea.