PREDICTION ERROR GUIDE
RMSE vs MAE
Both RMSE and MAE measure prediction error in the same units as the target, but RMSE gives more weight to large errors. This guide shows a worked example with and without an outlier.
Definitions
For n paired observations with actual y and predicted ŷ:
- errori = predictedi − actuali
- MSE = Σ(errori²) / n
- RMSE = √MSE
- MAE = Σ|errori| / n
RMSE squares each error before averaging, so a single large error can pull RMSE up sharply. MAE averages absolute errors, so every miss contributes linearly.
Worked example without an outlier
| Actual | Predicted | Error | |Error| | Error² |
|---|---|---|---|---|
| 3 | 2.5 | −0.5 | 0.5 | 0.25 |
| 5 | 5.2 | 0.2 | 0.2 | 0.04 |
| 2 | 2.1 | 0.1 | 0.1 | 0.01 |
| 7 | 6.3 | −0.7 | 0.7 | 0.49 |
- Σ|error| = 1.5; MAE = 1.5 / 4 = 0.375
- Σerror² = 0.79; MSE = 0.79 / 4 = 0.1975; RMSE = 0.444
RMSE and MAE are close because the errors are similar in size.
Same example with one large outlier
Replace the last row with a much larger prediction error:
| Actual | Predicted | Error | |Error| | Error² |
|---|---|---|---|---|
| 3 | 2.5 | −0.5 | 0.5 | 0.25 |
| 5 | 5.2 | 0.2 | 0.2 | 0.04 |
| 2 | 2.1 | 0.1 | 0.1 | 0.01 |
| 7 | 10.0 | 3.0 | 3.0 | 9.00 |
- Σ|error| = 3.8; MAE = 3.8 / 4 = 0.95 (about 2.5× larger)
- Σerror² = 9.30; MSE = 9.30 / 4 = 2.325; RMSE = 1.525 (about 3.4× larger)
RMSE grows faster because the large error is squared. Whether that is desirable depends on whether large misses are proportionally worse for your decision.
When to use which
- Use MAE when every unit of error matters equally. It is easy to interpret as the average miss.
- Use RMSE when large errors are especially harmful and you want to penalise them more heavily.
- Report both when possible; a big gap between them suggests a few large errors are dominating RMSE.
Try it
Paste either table into the RMSE calculator to reproduce the values.
Limitations
- Both metrics need the same units and the same evaluation rows for every model you compare.
- Neither shows whether errors are systematically high or low; inspect signed residuals.
- Neither accounts for uncertainty, data splitting or future observations.
- A small average error does not guarantee safe performance for every subgroup.