Why do you take the sqrt of 1/n for RMSE? Updated question:
Why do we use RMSE:
$$RMSE = \sqrt{\frac{1}{n}\Sigma_{i=1}^{n}{\Big(\hat{y}_i -y_i\Big)^2}}$$
Why is it not MRSE:
$$MRSE = \frac{1}{n}\sqrt{\Sigma_{i=1}^{n}{\Big(\hat{y}_i -y_i\Big)^2}}$$
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:
$$RMSE = \sqrt{\frac{1}{n}\Sigma_{i=1}^{n}{\Big(\hat{y}_i -y_i\Big)^2}}$$
Why is it not:
$$RMSE = \frac{1}{n}\sqrt{\Sigma_{i=1}^{n}{\Big(\hat{y}_i -y_i\Big)^2}}$$
What is the reason for taking the square root of 1/n?

I think both RMSE and MRSE could be potentially used for the purpose of creating a metric related to residuals per data point. The difference, and why the RMSE is commonly used and MRSE is not probably lies in interpretation of the terms and their related metrics.
If we roll back RMSE to SE at every step of the way we get a term that is commonly used and interpretable. We square RMSE and get MSE (variance), we square root it and get SE. However, rolling back MRSE does not give as nice a terms. We multiply MRSE by sample size and get RSE, which is not commonly used to evaluate anything. This may be the reason why one is used and the other isn't.
