Root Mean Squared Error (RMSE)

TL;DR

• Quantifies average prediction error in the original units of the target.
• Lower RMSE indicates predictions closer to actual values; higher RMSE indicates larger discrepancies.
• Computed by taking the square root of the average of squared differences between predictions and observations.

Definition

Root Mean Squared Error (RMSE) is the square root of the mean squared error (MSE), where MSE is the average of the squared differences between predicted values and actual values.

$$\mathrm{RMSE} = \sqrt{\frac{1}{n}\sum_{i=1}^n (\hat{y}_i - y_i)^2}$$

Explanation

RMSE measures the difference between predicted and actual values for a quantity. It is widely used to evaluate the performance of predictive models in fields such as statistics, machine learning, and engineering. RMSE is interpretable on the same scale as the target variable and is often reported alongside other metrics like mean absolute error (MAE) and R-squared to provide a fuller assessment of model accuracy.

A low RMSE indicates the model’s predictions are close to the actual values; a high RMSE indicates larger prediction errors.

Examples

Example 1: House prices

Suppose a predictive model estimates house prices from features such as size and location. With a dataset of 100 houses, the RMSE is calculated after making predictions:

RMSE = sqrt(mean((predicted_prices - actual_prices)^2))

A low RMSE means the model accurately predicts house prices; a high RMSE means there is a large difference between predicted and actual prices.

Example 2: Customer counts

Suppose a predictive model estimates the number of customers visiting a store each day. With a dataset of 100 days, the RMSE is calculated after making predictions:

RMSE = sqrt(mean((predicted_customer_count - actual_customer_count)^2))

A low RMSE means the model accurately predicts daily customer counts; a high RMSE means there is a large difference between predicted and actual numbers.

Use cases

• Evaluating predictive model performance in statistics, machine learning, and engineering.

Related terms

• Mean Absolute Error (MAE)
• R-squared (R²)