Home › AI / ML › ML › Regression Metrics

Regression Metrics: MAE, MSE, RMSE, R²

ML evaluation regression residuals metrics

It all starts with the residual

For regression, the raw material of every metric is the residual — the gap between what the model predicted and what actually happened.

residual = actual − predicted. A perfect model has all residuals at zero. Each metric is just a different way of summarising a whole pile of residuals into one number.

Build each metric from the errors

The animation takes five predictions, shows their residuals, then assembles MAE, MSE, RMSE, and finally R² one step at a time.

The four, compared

MAE mean |error|

Average absolute error, in the target's units. Easy to read, treats all misses equally, robust to outliers.

MSE mean error²

Squares each error first, so big misses dominate. Units are squared — harder to interpret directly.

RMSE √MSE

Square root of MSE — back in the target's units, but still penalises large errors heavily.

R² 0 → 1

Fraction of variance explained vs just predicting the mean. 1 = perfect, 0 = no better than the average.

MAE vs RMSE — which to report?

Prefer MAE when
  • You want errors in plain units, equally weighted
  • The data has outliers you don't want to dominate
  • A miss of 10 is simply "twice as bad" as a miss of 5
Prefer RMSE when
  • Large errors are disproportionately costly
  • You're optimising a model that minimises squared error
  • You want a metric sensitive to variance in errors
Reading R²

R² can be negative if the model is worse than guessing the mean. And a high R² doesn't guarantee good predictions — always look at RMSE/MAE in real units too.

Related

For classification, the metric family is different — see Confusion Matrix and Precision, Recall & F1. The residual idea here also underpins how Linear Regression is fit in the first place.