Difference between root mean square error and mean square error

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Root Mean Square Error (RMSE) vs. Mean Square Error (MSE): Key Differences

Definition and Calculation: RMSE and MSE

Root Mean Square Error (RMSE) and Mean Square Error (MSE) are both metrics used to measure the accuracy of models by quantifying the difference between predicted and observed values. MSE is calculated by taking the average of the squared differences between predictions and actual values. RMSE is simply the square root of the MSE, which brings the error metric back to the original units of the data, making it easier to interpret in practical terms Chai2014Chai2014Willmott2005.

Sensitivity to Error Distribution

RMSE is more sensitive to large errors than MSE because squaring the errors before averaging gives more weight to larger differences. This means that RMSE will increase more than MSE if there are a few large errors in the data. RMSE is particularly appropriate when the error distribution is expected to be Gaussian (normal), as it better reflects the impact of outliers in such cases Chai2014Chai2014Hodson2022+1 MORE.

Interpretability and Use Cases

MSE is useful for mathematical analysis and optimization because it is differentiable and works well in many statistical models. However, its unit is the square of the original data, which can make interpretation less intuitive. RMSE, by taking the square root, returns the error to the same unit as the original data, making it more interpretable for practical applications and easier to communicate to non-technical audiences Chai2014Chai2014Willmott2005.

Statistical Properties and Model Evaluation

RMSE satisfies the triangle inequality, making it a true distance metric, while MSE does not. This property can be important in certain mathematical contexts. RMSE is often preferred when comparing model performance, especially when the error distribution is normal or when large errors are particularly undesirable. However, both RMSE and MSE can be misleading if used alone, especially when the error distribution is not normal or contains outliers. In such cases, it is recommended to use a combination of metrics for a more complete assessment Chai2014Chai2014Willmott2005+2 MORE.

Practical Recommendations

Use RMSE when you want an error metric in the same units as your data and when large errors are especially important to penalize.
Use MSE for mathematical convenience, especially in optimization and statistical modeling.
Consider the distribution of your errors: RMSE is optimal for normal distributions, while other metrics may be better for non-normal error distributions Chai2014Chai2014Hodson2022+1 MORE.

Conclusion

The main difference between RMSE and MSE is that RMSE is the square root of MSE, making it more interpretable and sensitive to large errors. Both metrics have their strengths and are best used in combination with other evaluation tools, depending on the error distribution and the specific goals of the analysis Chai2014Chai2014Willmott2005+2 MORE.

Sources and full results

Most relevant research papers on this topic

Root mean square error (RMSE) or mean absolute error (MAE)? – Arguments against avoiding RMSE in the literature

Using a combination of RMSE and MAE metrics in model evaluation is often more beneficial than avoiding either metric alone, as RMSE is more appropriate for Gaussian error distributions and satisfies the triangle inequality requirement.

Highly Cited

2014·

5452citations

·T. Chai et al.

Geoscientific Model Development·

DOI

Root mean square error (RMSE) or mean absolute error (MAE)

The root mean square error (RMSE) is more appropriate for representing model performance than the mean absolute error (MAE) when the error distribution is expected to be Gaussian, but a combination of metrics is often necessary to assess model performance.

Highly Cited

2014·

894citations

·T. Chai et al.

Geoscientific Model Development Discussions·

DOI

Advantages of the mean absolute error (MAE) over the root mean square error (RMSE) in assessing average model performance

The mean absolute error (MAE) is a more natural and unambiguous measure of average model-performance error than the root mean square error (RMSE), making it a better choice for dimensional evaluations and inter-comparisons.

Highly Cited

2005·

5664citations

·C. Willmott et al.

Climate Research·

DOI

Root-mean-square error (RMSE) or mean absolute error (MAE): when to use them or not

Both RMSE and MAE are useful for evaluating models, but their optimal use depends on the type of error and the distribution of the error distribution.

Highly Cited

2022·

1938citations

·T. Hodson

Geoscientific Model Development·

DOI

Root mean square error or mean absolute error? Use their ratio as well

The RMSE estimator is preferred for platykurtic distributions, MAE for leptokurtic distributions, and either RMSE or MAE for mesokurtic distributions, while their ratio can identify errors from platykurtic, mesokurtic, and leptokurtic distributions

Highly Cited

2021·

607citations

·D. Karunasingha

Inf. Sci.·

DOI

Estimating the uncertainty in estimates of root mean square error of prediction: application to determining the size of an adequate test set in multivariate calibration

The two methods for estimating uncertainty in root mean square error of prediction (RMSEP) agree closely, and using a correction factor can ensure accurate results in multivariate calibration.

Highly Cited

1999·

114citations

·N. Faber

Chemometrics and Intelligent Laboratory Systems·

DOI

Decomposition of the mean absolute error (MAE) into systematic and unsystematic components

This study presents a three-part decomposition of mean absolute error (MAE) into bias error, proportionality error, and unsystematic error, providing more straightforward information on the nature of model-error distributions.

Highly Cited

2023·

171citations

·S. Robeson et al.

PLOS ONE·

DOI

Direct Root-Mean-Square Error for Surface Accuracy Evaluation of Large Deployable Mesh Reflectors

Direct root-mean-square error and effective region root-mean-square error provide more accurate and efficient surface accuracy evaluations for large deployable mesh reflectors.

2020·

13citations

·S. Yuan et al.

DOI

Mean Square Error of Prediction as a Criterion for Selecting Variables

The mean square error of prediction is a more meaningful criterion for selecting variables than the residual sum of squares, as it utilizes the values of predictor variables and the magnitude of the estimated variance.

Highly Cited

1971·

651citations

·D. M. Allen

Technometrics·

DOI

Soundness and completeness of quantum root-mean-square errors

This paper proposes an improved quantum generalization of the classical root-mean-square error, which is state-dependent and operationally definable, and perfectly characterizes precise measurements without changing previous uncertainty relations.

Highly Cited

2018·

231citations

·M. Ozawa

npj Quantum Information·

DOI

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