📋

Walk-through

How to Use This Calculator

3 steps

1

Enter your actual and predicted values

List the real, observed values in the Actual field and your model's forecasts or predictions in the Predicted field, in matching order. Both lists must contain the same number of values — the calculator pairs them position by position.

2

Read MSE, RMSE, and MAE

The MSE tab shows the mean squared error and a row-by-row table of residuals and squared errors. Switch to the RMSE tab to see the error in your data's original units, or the MAE tab to see the average absolute error, which is less sensitive to outliers.

3

Compare models or scenarios

Swap in a different set of predictions (from another model, or a revised forecast) to see how MSE, RMSE, and MAE change. A lower value on all three means the new predictions fit the actual data more closely.

⚡

Reference

Formula & Methodology

3 formulas

Mean squared error

MSE = Σ(actual − predicted)² / n

MSE averages the squared difference between each actual value and its corresponding predicted value across all n paired observations. Squaring the residual makes every error positive and penalizes large errors more heavily than small ones.

Root mean squared error

RMSE = √MSE

Taking the square root of MSE returns the error to the same units as the original data, which makes RMSE easier to interpret directly against your actual values than the squared MSE.

Mean absolute error

MAE = Σ|actual − predicted| / n

MAE averages the absolute value of each residual instead of squaring it, so every error contributes proportionally to its size rather than its square — large outliers have less influence on MAE than on MSE or RMSE.

📖

Glossary

Key Terms Explained

7 terms

Mean squared error (MSE) The average of the squared differences between actual and predicted values. Expressed in squared units of the original data, MSE is the standard loss function minimized by ordinary least-squares regression.

Root mean squared error (RMSE) The square root of MSE. RMSE is expressed in the same units as the original data, making it more directly interpretable than MSE while still penalizing large errors more than small ones.

Mean absolute error (MAE) The average of the absolute (unsquared) differences between actual and predicted values. MAE is more robust to outliers than MSE or RMSE because it does not square the residuals.

Prediction error The general term for the difference between what a model predicted and what actually occurred. MSE, RMSE, and MAE are all summary statistics that aggregate prediction error across many observations.

Residual The signed difference between a single actual value and its corresponding predicted value (actual − predicted). Residuals are the building blocks of MSE, RMSE, and MAE.

Regression A modeling technique that predicts a continuous numeric outcome from one or more input variables. MSE is the loss function most commonly minimized when fitting a regression model.

Forecast accuracy How closely a model's predicted values match the actual outcomes that later occurred. MSE, RMSE, and MAE are three of the most common metrics used to quantify forecast accuracy.

👥

Scenarios

Real-World Examples

3 worked examples

📈

Regression model check

Small validation set

Actual 3, -0.5, 2, 7 Predicted 2.5, 0, 2, 8

The four residuals are 0.5, -0.5, 0, and -1. Squaring and averaging them gives an MSE of 0.375, so RMSE ≈ 0.612 — meaning predictions are typically off by about 0.61 units. MAE is 0.5, since the average absolute error is smaller than the RMSE would suggest, showing the errors are fairly uniform rather than dominated by one large miss.

📊

Sales forecast

Five-point forecast vs. actuals

Actual 10, 20, 30, 40, 50 Predicted 12, 18, 33, 36, 52

Errors of -2, 2, -3, 4, and -2 units square to 4, 4, 9, 16, and 4, giving an MSE of 7.4 and an RMSE of about 2.72 units. The point predicted at 36 (actual 40) contributes the largest squared error, so it drives RMSE up more than MAE (which averages the absolute errors evenly at 2.6).

🎯

Outlier stress test

One large miss among otherwise close predictions

Actual 100, 150, 200, 250, 300 Predicted 110, 140, 210, 240, 305

Even though four of the five predictions are within 10 units, MSE and RMSE weight the largest residual (250 vs. 240, a miss of 10) the same as the others here because all residuals are similar size — but if one residual were much larger, MSE/RMSE would rise sharply while MAE would rise only modestly, illustrating why MAE is the more outlier-resistant metric of the two.

📄

Deep Dive

Understanding Mean Squared Error, RMSE, and MAE

3 chapters · 4 min read

Mean squared error (MSE) is the most widely used metric for scoring how well a regression or forecasting model's predictions match reality. This calculator walks through MSE alongside its two closest relatives — RMSE and MAE — so you can see not just the number but what it means for your model.

How MSE, RMSE, and MAE work

MSE = Σ(actual − predicted)² / n squares every residual before averaging, which means a single large miss can dominate the result — a residual twice as large contributes four times the squared error. RMSE = √MSE undoes the squaring at the end, returning the metric to the same units as your original data so it is directly comparable to the values you are predicting. MAE = Σ|actual − predicted| / n skips squaring entirely and averages the absolute size of each error, so it treats a miss of 10 as exactly twice as bad as a miss of 5 — no more, no less.

Inputs and what they mean

Enter your Actual values (the real, observed outcomes) and Predicted values (what a model or forecast produced) as two matching lists, separated by commas, spaces, or new lines. Order matters: the first actual value is paired with the first predicted value, and so on. Both lists must contain the same number of entries — the calculator flags a mismatch rather than guessing which values pair together.

Limits and edge cases

MSE and RMSE are sensitive to outliers: a single very large residual can inflate both metrics even if every other prediction is accurate, which is sometimes exactly what you want (when large errors are especially costly) and sometimes misleading (when one bad data point skews the whole score). MAE is more robust to outliers but treats all error sizes proportionally, so it can undersell the impact of a rare, severe miss. None of these three metrics alone tells you whether a model is statistically significant or will generalize to new data — for that, pair them with out-of-sample testing, a coefficient of determination (R²), or cross-validation.

❓

Questions

Frequently Asked Questions

6 questions

What is the formula for mean squared error?+

MSE = Σ(actual − predicted)² / n. Take the difference between each actual and predicted value, square it, sum all the squared differences, and divide by the number of paired observations (n).

How do I calculate RMSE from MSE?+

RMSE is simply the square root of MSE: RMSE = √MSE. This calculator computes both automatically once you enter matching actual and predicted values.

What's the difference between MSE/RMSE and MAE?+

MSE and RMSE square each residual before averaging, so large errors are penalized disproportionately more than small ones. MAE averages the absolute value of each residual instead, treating every unit of error the same size regardless of how large the individual miss was — making MAE more resistant to outliers.

What units are MSE, RMSE, and MAE expressed in?+

MSE is expressed in squared units of your original data (for example, dollars² if you're predicting dollars), which makes it hard to interpret directly. RMSE and MAE are both expressed in the same units as your original data, so they can be compared directly against the actual values.

Is a lower MSE always better?+

Yes — for a given data set and modeling task, a lower MSE (and correspondingly lower RMSE and MAE) indicates predictions that are closer to the actual values on average. However, comparing MSE across different data sets or different scales of data isn't meaningful, since MSE depends on the magnitude of the values being predicted.

How are MSE, RMSE, and MAE used in regression?+

MSE is the loss function that ordinary least-squares regression minimizes directly, which is why it's often reported alongside a fitted regression model. RMSE and MAE are commonly used afterward, on a held-out validation or test set, to report how accurately the model performs on data it wasn't trained on.

Download a one-page PDF of your numbers instantly. Add your email to also get our occasional calculator tips — no spam, unsubscribe anytime. Privacy.

Mean Squared Error Calculator

Actual vs. Predicted Values