How to use it
- Paste the actual (observed) values into the first box.
- Paste the predicted values, in the same order, into the second box.
- Read RMSE, MAE and the other measures.
Formulas
With n pairs and error eᵢ = actualᵢ − predictedᵢ:
- MAE = Σ|eᵢ| ÷ n
- MSE = Σeᵢ² ÷ n
- RMSE = √MSE
- MAPE = 100 × Σ|eᵢ ÷ actualᵢ| ÷ n, in percent
- Bias = Σ(predictedᵢ − actualᵢ) ÷ n; positive means the predictions are too high on average
- R² = 1 − Σeᵢ² ÷ Σ(actualᵢ − mean of actual)²
Worked example
Actual: 3, −0.5, 2, 7. Predicted: 2.5, 0, 2, 8.
Errors: 0.5, −0.5, 0, −1. MAE = (0.5 + 0.5 + 0 + 1) ÷ 4 = 0.5. MSE = (0.25 + 0.25 + 0 + 1) ÷ 4 = 0.375, so RMSE = √0.375 = 0.612. MAPE = (0.5/3 + 0.5/0.5 + 0 + 1/7) ÷ 4 × 100 = 32.7%; the value −0.5 alone contributes 100%, which shows how MAPE behaves near zero. Bias = (−0.5 + 0.5 + 0 + 1) ÷ 4 = 0.25. The mean of the actual values is 2.875 and Σ(actual − mean)² = 29.19, so R² = 1 − 1.5 ÷ 29.19 = 0.949.
Tips and common mistakes
- Keep the two lists aligned: one missing value shifts every pair after it.
- Compare RMSE only between models predicting the same data in the same units.
- MSE is in squared units; report RMSE when you want a number in the data’s own units.