Enter your observed values and predicted (simulated) values as two columns of numbers, and this RMSE Calculator computes the Root Mean Square Error between them. Paste or type your data pairs — one per line — and get the RMSE, mean error, and count of data points back immediately.
Root Mean Square Error (RMSE)
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Mean Square Error (MSE)
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Mean Absolute Error (MAE)
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Mean Error (Bias)
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Number of Data Pairs
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RMSE is a standard statistical metric that measures how far predicted values deviate from observed (actual) values. It is calculated by taking the square root of the average of squared differences between predictions and observations. Lower RMSE values indicate a better fit between your model and the data.
RMSE = √[ (1/n) × Σ(observedᵢ − predictedᵢ)² ]. You subtract each predicted value from the corresponding observed value, square the result, average all squared differences, and then take the square root of that average.
MSE (Mean Square Error) is the average of the squared errors without taking the square root. RMSE is simply the square root of MSE, which brings the error metric back to the same units as your original data. RMSE is therefore easier to interpret in context.
There is no universal 'good' RMSE threshold — it depends on the scale of your data. A useful approach is to compare RMSE to the range or standard deviation of your observed values. An RMSE that is small relative to the spread of your data generally indicates a good model.
Both measure average prediction error, but RMSE penalises large errors more heavily because it squares each difference before averaging. MAE (Mean Absolute Error) treats all errors equally. If large outliers are especially costly in your application, RMSE is the more sensitive measure.
Type or paste your observed values into the left text box and your predicted (simulated) values into the right text box, with one number per line. Both columns must have the same number of values. The calculator will automatically pair them row by row.
No. Because RMSE is derived from squared differences, it is always zero or positive. An RMSE of zero would mean your predictions perfectly match all observed values.
Mean Error (also called bias) is the average of (observed − predicted) without squaring. A positive mean error means your model tends to underpredict; a negative mean error indicates overprediction. Unlike RMSE, positive and negative errors can cancel out, so mean error alone does not capture overall accuracy.