Step by Step
MAE
Mean Absolute Error — robust to outliers, original units
The average of the absolute differences between predictions and actual values, expressed in the original units of the target variable. Because it doesn't square errors, it's more robust to outliers than MSE.
Example: an MAE of $15,000 for house price predictions means predictions are, on average, off by $15,000 in either direction.
MSE
Mean Squared Error — penalizes large errors, not original units
The average of the squared differences between predictions and actual values. Squaring disproportionately penalizes large errors, but the result is no longer in the original units of the target variable.
Example: a single wildly inaccurate prediction contributes disproportionately more to MSE than to MAE, since the error gets squared.
RMSE
Root Mean Squared Error — back in original units, most commonly reported
The square root of MSE, bringing the metric back into the original units of the target variable while still penalizing large errors more than MAE does — this is the most commonly reported regression metric in practice.
Example: an RMSE of $18,000 for house price predictions is directly interpretable in dollars, unlike raw MSE.
R²
R-squared — proportion of variance explained, can be negative
Represents the proportion of variance in the target variable explained by the model. An R² of 1.0 is perfect, 0 means the model is no better than simply predicting the mean every time, and it CAN be negative if the model performs worse than that naive baseline.
Example: an R² of -0.2 would mean the model is actually performing worse than simply predicting the average house price for every single house, regardless of its actual features.
Applied Walkthrough
1
A regression model's R² is calculated and comes out negative.
2
Ask: what does a negative R² actually indicate? That the model performs worse than the simplest possible baseline — just predicting the mean value for every example.
3
This is a serious red flag, since R² of exactly 0 already represents this naive "predict the mean" baseline — going negative means the model is actively worse than doing nothing sophisticated at all.
4
The team should investigate this model further, since a negative R² suggests something has likely gone wrong (overfitting, a bug, or a fundamentally poor model choice) rather than the model simply being "mediocre."
Exam Application
Exams test whether you can distinguish these four regression metrics by their key properties: MAE (robust to outliers, original units), MSE (penalizes large errors, not original units), RMSE (original units, most commonly reported), and R² (proportion of variance explained, can be negative). Expect a specific test of whether you understand what a negative R² means.
⚠ Common Trap
The most common trap is assuming R² can only range from 0 to 1, treating a negative value as impossible or a calculation error. R² CAN legitimately be negative, and it specifically signals that a model is performing worse than the naive baseline of just predicting the mean for every example.
✓ Quick Self-Check
1. Which regression metric is most robust to outliers, and why?
MAE, because it uses absolute differences rather than squared differences, avoiding disproportionate penalties for large errors.
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2. Which regression metric penalizes large errors most heavily, but isn't in the original units?
MSE.
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3. Why is RMSE more commonly reported than raw MSE?
Because taking the square root brings the metric back into the original units of the target variable, making it more directly interpretable.
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4. What does an R² of exactly 0 represent?
A model that performs no better than simply predicting the mean value for every example.
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5. Can R² be negative, and if so, what does that indicate?
Yes — a negative R² indicates the model performs worse than the naive baseline of predicting the mean for every example.
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