About the MSE, RMSE & MAE Calculator
This mean squared error calculator measures how far a set of predictions is from the actual values. Paste the observed values in one box and the forecasts or model predictions in the other, in the same order, and it returns the mean squared error (MSE), root mean squared error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), the mean error (bias), the sum of squared errors and R².
It is useful for data science and machine learning students comparing regression models, demand planners checking last quarter's forecast, and analysts reporting forecast accuracy to a manager. A table lists the error, squared error and absolute error for every point, and a chart shows actual against predicted so outliers jump out.
Errors are defined as actual minus predicted, so a positive mean error means the model under-predicts on average. MSE here divides by n (the number of pairs), which is the standard definition for evaluating predictions; it does not subtract model degrees of freedom the way a regression's residual variance does. MAPE skips any pair whose actual value is 0.
With the default inputs, the mean squared error (mse) is 11. Change any value above to recalculate instantly.
How to use the mse, rmse & mae calculator
- 1Paste the actual (observed) values into the first box.
- 2Paste the predicted values into the second box in the same order.
- 3Read MSE, RMSE and MAE in the results.
- 4Check MAPE and bias to see relative accuracy and any systematic over- or under-prediction.
- 5Use the error table and chart to spot the points with the biggest misses.
Formula and method
Each error is the actual value minus the prediction. MSE squares every error before averaging, so large misses count much more than small ones and the result is in squared units. RMSE takes the square root to bring the metric back to the original units, which makes it easier to interpret. MAE averages the absolute errors and treats every unit of error equally, so it is less sensitive to outliers.
MAPE expresses the average absolute error as a percentage of the actual value (pairs with an actual of zero are excluded because the percentage is undefined). The mean error shows systematic bias — positive values mean the predictions are too low on average. R² compares the squared error with the total variation of the actual values around their mean; it can be negative when the predictions are worse than simply predicting the mean.
- yᵢ
- Actual (observed) value
- ŷᵢ
- Predicted (forecast) value
- eᵢ
- Error, actual minus predicted
- n
- Number of actual–predicted pairs
- SSE, SST
- Sum of squared errors; total sum of squares of actual values about their mean
Worked examples
Monthly sales forecast vs actual (8 months)
The errors are 2, −2, 4, −2, −4, 2, 6 and −2. Their squares add up to 88, so MSE = 88 ÷ 8 = 11 and RMSE = √11 ≈ 3.32. The absolute errors average 3, and the forecast is too low by 0.5 units on average.
Model with one big miss
Errors are −1, 0, 1, 0 and −8. One large miss makes the squared errors sum to 66, so MSE = 13.2 and RMSE ≈ 3.63, while MAE is only 2. R² is negative (−0.65) because predicting the mean of 14 every time would have had a smaller squared error (SST = 40).
Weekly demand with a zero-sales week (MAPE skips it)
Errors are −1, 1, −2 and 0, so SSE = 6, MSE = 1.5 and RMSE ≈ 1.22. The first week has actual sales of 0, so MAPE averages only the other three percentage errors: (20% + 25% + 0%) ÷ 3 = 15%. With SST = 32.75, R² = 1 − 6/32.75 ≈ 0.817.
Frequently asked questions
What is the difference between MSE, RMSE and MAE?+
MSE averages the squared errors, RMSE is its square root (back in the original units), and MAE averages the absolute errors. RMSE and MSE penalise large errors more heavily; MAE weights every unit of error equally and is more robust to outliers.
What is a good RMSE value?+
There is no universal threshold because RMSE is in the units of your data. Compare it with the scale of the values (for example RMSE divided by the mean), with a naive benchmark such as last period's value, or with competing models on the same data.
Why is RMSE always greater than or equal to MAE?+
Squaring before averaging gives large errors more weight, and the root of a mean square is never less than the mean of absolute values. They are equal only when every error has the same magnitude; a big gap signals a few large errors.
Should MSE divide by n or n − p?+
For judging predictions, divide by n, as this calculator does. Inside a regression, the residual variance divides by n − p (the number of fitted parameters) to give an unbiased estimate of the error variance, so the two numbers differ slightly.
Why is MAPE undefined when an actual value is zero?+
MAPE divides each error by the actual value, and dividing by zero is undefined. Pairs with an actual of 0 are skipped. If many actuals are zero or near zero, use MAE or a scaled metric like MASE instead.