How to use this calculator
- Enter or choose actual (observed) values.
- Enter or choose predicted (forecast) values, same order.
- The rMSE updates as you type. Use Copy result or Share to keep it.
Method
eᵢ = actualᵢ − predictedᵢ; MSE = Σeᵢ² ÷ n; RMSE = √MSE
- eᵢ
- error for pair i
- n
- number of pairs (the divisor is n, not n − 1 or n − k)
Worked example
- Errors: 0.5, −0.5, 0, 1. Squares: 0.25, 0.25, 0, 1.
- MSE = 1.5 ÷ 4 = 0.375.
- RMSE = √0.375 = 0.6124.
The predictions are off by about 0.61 units in the root-mean-square sense, slightly more than the 0.5 mean absolute error because of the one larger miss.
What the result means
RMSE is always at least as large as MAE; a big gap between them means a few large errors. Compare RMSE only between models predicting the same quantity in the same units, or use the normalised versions. A non-zero mean error shows systematic over- or under-prediction.
Assumptions and limits
- Each predicted value is paired with the actual value in the same position.
- RMSE here divides by n (descriptive). The residual standard error of a regression divides by n − k − 1 instead.
- Sensitive to outliers by design; report MAE alongside it if a few large errors should not dominate.
Questions
RMSE or MAE?
RMSE penalises large errors more, which suits costs that grow faster than the error. MAE treats every unit of error equally and is easier to explain.
Is a lower RMSE always better?
For the same data, yes. Across different data sets or units the raw numbers are not comparable.