Regression metrics: MAE, RMSE, R²

Three measures of «how wrong does the model guess?» when the answer is a number:

Metric	Formula	Property
MAE	the mean of |y − ŷ|	in the same unit, every error weighs the same
RMSE	√(the mean of (y − ŷ)²)	in the same unit, punishes large errors hard
R²	1 − SS_res/SS_tot	the share of the variation that is explained; 1 = perfect, 0 = as good as always guessing the mean

RMSE ≥ MAE always. If RMSE is much larger than MAE there are a few large errors — go and look at them.

R² can go negative. That means the model is worse than always guessing the mean — which happens more often than you would think on test data.

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

y    = np.array([100, 200, 300, 400, 500])
pred = np.array([110, 190, 310, 380, 700])       # the last one is badly wrong

print(round(mean_absolute_error(y, pred), 1))                      # 50.0
print(round(mean_squared_error(y, pred) ** 0.5, 1))                # 92.2  ← pulled up by the 200 error
print(round(r2_score(y, pred), 3))                                 # 0.575

# without the outlier:
print(round(mean_absolute_error(y[:4], pred[:4]), 1),
      round(mean_squared_error(y[:4], pred[:4]) ** 0.5, 1))        # 12.5 14.6

Choosing the metric: MAE when every error costs the same per unit (delivery time in days). RMSE when large errors are disproportionately expensive (electricity grid load). And always report a baseline — R² does that for you by comparing against the mean guess.