RMSE: What does it mean?

Contributed by: Shweta Gupta

Root-Mean-Square-Error or RMSE is one of the most popular measures to estimate the accuracy of our forecasting model’s predicted values versus the actual or observed values while training the regression models or time series models.

It measures the error in our predicted values when the target or response variable is a continuous number. For example, when using regression models to predict a quantity like income, sales value/volumes, demand volumes, scores, height or weight etc.

Thus, RMSE is a standard deviation of prediction errors or residuals. It indicates how spread out the data is around the line of best fit.

It is also an essential criterion in shortlisting the best performing model among different forecasting models that you may have trained on one particular dataset. To do so, simply compare the RMSE values across all models and select the one with the lowest value on RMSE.

Such a shortlisted model produces the lowest error in predicting values for the Target variable.

RMSE also has the useful property of being on the same scale/units as the Target variable. Hence it is very intuitive to understand as well.

But how exactly is this measure calculated?

As the name suggests, it is the square root of average squared errors between observed and predicted values for the target variable.

Therefore, to calculate RMSE, the formula is as follows:

Where:

• ∑ is the summation of all values
• f is the predicted value
• o is observed or actual value
• (fi — oi) 2 are the differences between predicted and observed values and squared
• N is the total sample size

Although the above formula may look a bit daunting, all it is doing is simplified in below steps:

1. For every predicted value, calculate the difference from corresponding observed value
2. Square the difference arrived at step 1. Repeat for all differences i.e (Predicted — Observed)2 for every observation in the sample
3. Sum all the “squared differences” calculated in step 2
4. Calculate the average of “sum of squared differences” derived in step 3. This value is called MSE or Mean Squared Error
5. Finally take the square root of the value derived in step 4. This value is RMSE

Therefore, to summarize our learnings on RMSE:

• RMSE is the standard deviation of the residuals
• RMSE indicates average model prediction error
• The lower values indicate a better fit
• It is measured in same units as the Target variable