RMSE (Root Mean Square Error) is the square root of the average of squared differences between observed and predicted values. It expresses prediction error in the same units as the original data, making it easy to interpret alongside your measurements.

What is the formula for RMSE?

RMSE = sqrt( (1/n) * sum( (observed_i - predicted_i)^2 ) ). First compute the squared error for each pair, average those squared errors (MSE), then take the square root.

How is RMSE different from MAE?

MAE (Mean Absolute Error) treats all errors equally. RMSE squares each error before averaging, so large errors are penalized more heavily. If large prediction mistakes are particularly costly, RMSE is more informative; if all errors should be weighted equally, use MAE.

What is a good RMSE value?

There is no universal threshold. RMSE must be interpreted relative to the scale of your data and domain. A common rule of thumb is to compare RMSE to the standard deviation of the observed values — an RMSE much smaller than the standard deviation indicates a model that adds predictive value.

Can RMSE be negative?

No. RMSE is always zero or positive because it is the square root of a sum of squared (non-negative) values. RMSE equals zero only when every prediction is exactly correct.

MSE (Mean Squared Error) is the average of squared errors — it is RMSE squared. MSE penalizes large errors even more heavily than RMSE, but its value is in squared units of the original data, which can make physical interpretation harder.

RMSE Calculator

Enter observed and predicted value pairs to instantly compute RMSE, MSE, and MAE with a per-row squared error breakdown.

Bulk Import (optional)

Paste two columns of numbers separated by commas, one pair per line. Example: 3.5, 3.2

Data Pairs

# Observed Predicted

Results

RMSE —

MSE —

MAE —

Enter at least 1 valid pair to compute metrics.

Per-Row Squared Error Breakdown

#	Observed	Predicted	Error	Sq. Error	% of MSE
No data yet

Summary

Root Mean Square Error (RMSE) is one of the most widely used metrics for evaluating the accuracy of regression models and numerical predictions. It measures the average magnitude of prediction errors in the same units as the original data, giving more weight to large errors than Mean Absolute Error (MAE) does. This calculator accepts any number of observed-predicted pairs and returns RMSE, MSE, and MAE along with a per-row squared error table so you can spot outlier predictions immediately.

How it works

Enter your observed (actual) and predicted values — one pair per row.
Use "Add Row" to add more data pairs or paste comma-separated values into the bulk import box.
The tool computes the squared error for each row and displays a color-coded breakdown table.
RMSE, MSE, and MAE are recalculated instantly as you type.
Click "Copy Results" to copy the summary metrics to your clipboard.
Use "Reset" to clear all rows and start a new calculation.

Use cases

Evaluate regression model accuracy in machine learning and data science.
Compare forecast error between different models or time periods.
Assess sensor measurement accuracy against a known reference.
Calculate prediction error for financial or demand forecasting models.
Validate simulation output against observed experimental data.
Report model performance metrics in research papers or reports.
Identify which individual predictions contribute most to overall error.

Frequently Asked Questions

Last updated: 2026-06-11 · Reviewed by Nham Vu