# RMSE Calculator

> Paste actual and predicted values to get the root mean square error, MSE, MAE, mean bias and RMSE normalised by the mean and by the range.

Category: Statistics & probability · Applies to: Worldwide · Last updated: 2026-10-29
Interactive calculator: https://calculators.geekmidst.com/calculators/root-mean-square-error-calculator

RMSE is the typical size of a prediction error in the units of the data, with large errors weighted more heavily because they are squared before averaging.

Also called: Root mean squared error calculator, MSE calculator.

## Inputs
- Actual (observed) values (default 3, 5, 7, 9)
- Predicted (forecast) values, same order (default 2.5, 5.5, 7, 8)

## Results
- RMSE
- MSE (mean squared error)
- MAE (mean absolute error)
- Mean error (actual − predicted)
- RMSE ÷ mean of actual values
- RMSE ÷ range of actual values
- Pairs (n)

## Formula

```
eᵢ = actualᵢ − predictedᵢ;  MSE = Σeᵢ² ÷ n;  RMSE = √MSE
```

Where:
- eᵢ: error for pair i
- n: number of pairs (the divisor is n, not n − 1 or n − k)

## Worked example
1. Errors: 0.5, −0.5, 0, 1. Squares: 0.25, 0.25, 0, 1.
2. MSE = 1.5 ÷ 4 = 0.375.
3. RMSE = √0.375 = 0.6124.

**Result:** 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.

## Related calculators
- [Mean Absolute Error Calculator](https://calculators.geekmidst.com/calculators/mean-absolute-error-calculator)
- [MAPE Calculator](https://calculators.geekmidst.com/calculators/mean-absolute-percentage-error-calculator)
- [R-Squared Calculator](https://calculators.geekmidst.com/calculators/regression-r-squared-calculator)
- [Standard Deviation Calculator](https://calculators.geekmidst.com/calculators/standard-deviation-calculator)

Source: GM¹X CalcVerse — https://calculators.geekmidst.com/calculators/root-mean-square-error-calculator