Regression metrics measure how far predictions are from the true values. They differ in how they punish large errors, in their units, and in what baseline they compare against.

Metric	Formula	Units	Notes
MAE	`mean \|y − ŷ\|`	Target's	Every unit of error counts the same; less sensitive to outliers than MSE
MSE	`mean (y − ŷ)²`	Target's squared	Punishes large errors heavily; the usual training loss
RMSE	`√MSE`	Target's	Like MSE but interpretable; always ≥ MAE
R²	`1 − SS_res / SS_tot`	Unitless	Share of variance explained relative to predicting the mean
Adjusted R²	`1 − (1 − R²)(n − 1)/(n − p − 1)`	Unitless	Penalises extra predictors
MAPE	`mean \|y − ŷ\| / \|y\|`	%	Scale-free, but undefined at y = 0, explodes near 0, asymmetric
RMSLE	RMSE of `log(1 + y)`	≈ relative	For targets spanning orders of magnitude; penalises under-prediction more
Median absolute error	`median \|y − ŷ\|`	Target's	Robust to a few huge errors

SS_res = Σ(yᵢ − ŷᵢ)² SS_tot = Σ(yᵢ − ȳ)² R² = 1 − SS_res / SS_tot

Python

import numpy as np
from sklearn.metrics import mean_absolute_error, mean_squared_error, r2_score, root_mean_squared_error

y = np.array([100, 150, 200, 250, 300])
y_hat = np.array([110, 140, 210, 240, 380])            # one big miss

print(mean_absolute_error(y, y_hat), mean_squared_error(y, y_hat))    # 24.0 1360.0
print(round(root_mean_squared_error(y, y_hat), 2))                    # 36.88 ← pulled up by the 80 error
ss_res, ss_tot = np.sum((y - y_hat) ** 2), np.sum((y - y.mean()) ** 2)
print(round(1 - ss_res / ss_tot, 4), round(r2_score(y, y_hat), 4))    # 0.728 0.728

R² can be negative

R² compares the model with always predicting the mean of the true values. A model worse than that baseline has R² < 0 — common on a test set when a model has overfit or the data has drifted. R² = 1 is perfect; there is no lower bound.

Python

import numpy as np
from sklearn.metrics import r2_score

y = np.array([3.0, 5.0, 7.0, 9.0])
print(r2_score(y, np.full(4, y.mean())))              # 0.0 ← predicting the mean
print(r2_score(y, np.array([9.0, 7.0, 5.0, 3.0])))    # -3.0 ← worse than the mean

MAE vs RMSE, and the MAPE trap

Python

import numpy as np
from sklearn.metrics import mean_absolute_error, mean_absolute_percentage_error, root_mean_squared_error

y = np.array([10.0, 10.0, 10.0, 10.0])
spread = np.array([12.0, 8.0, 12.0, 8.0])      # four errors of 2
one_big = np.array([10.0, 10.0, 10.0, 18.0])   # one error of 8
print(mean_absolute_error(y, spread), mean_absolute_error(y, one_big))            # 2.0 2.0 ← MAE: same
print(root_mean_squared_error(y, spread), root_mean_squared_error(y, one_big))    # 2.0 4.0 ← RMSE prefers spread-out errors

demand = np.array([0.5, 100.0])                 # a near-zero actual
print(round(mean_absolute_percentage_error(demand, np.array([1.5, 100.0])), 2))   # 1.0 ← off by 1 unit → 100% MAPE

Choose	When
MAE	Errors cost proportionally; outliers shouldn't dominate; you want the median-type prediction
RMSE	Large errors are disproportionately bad; matches a squared-error training loss
R²	Communicating fit relative to a naive baseline (always report an error metric too)
MAPE / sMAPE / WAPE	Stakeholders think in percentages and actuals are far from 0 (WAPE = Σ\|e\| / Σ\|y\| is more stable)
RMSLE	Targets such as prices or counts spanning orders of magnitude

Can R² be negative?

MAE or RMSE?