# Solved – Why do you take the sqrt of 1/n for RMSE

Updated question:

Why do we use RMSE:
$$RMSE = sqrt{frac{1}{n}Sigma_{i=1}^{n}{Big(hat{y}_i -y_iBig)^2}}$$

Why is it not MRSE:
$$MRSE = frac{1}{n}sqrt{Sigma_{i=1}^{n}{Big(hat{y}_i -y_iBig)^2}}$$

I understand that other methods (e.g., MAE and MAPE) can be used as a metric for error. My question is specifically about why we use RMSE over MRSE.

Original:

Why is the equation for RMSE:
$$RMSE = sqrt{frac{1}{n}Sigma_{i=1}^{n}{Big(hat{y}_i -y_iBig)^2}}$$

Why is it not:
$$RMSE = frac{1}{n}sqrt{Sigma_{i=1}^{n}{Big(hat{y}_i -y_iBig)^2}}$$

What is the reason for taking the square root of 1/n?

Contents

While Demetri's answer gives a very good derivation or RMSE, it doesn't really explain why not the other method you suggest. I think you can get a little more insight by observing that MRSE is not a valid name for your suggested measure. Look closely and the steps are

1. Square the residuals
$$MRSE = frac{1}{n} sum sqrt{(hat{y}_i – y_i)^2} = frac{1}{n}sum |hat{y}_i – y_i| = MAE$$