To evaluate trained models for usefulness / accuracy.
To serve as a heuristic for training models. In data science, RMSE has a double purpose: RMSE in Data Science: Subtleties of Using RMSE If the noise is small, as estimated by RMSE, this generally means our model is good at predicting our observed data, and if RMSE is large, this generally means our model is failing to account for important features underlying our data. The random noise here could be anything that our model does not capture (e.g., unknown variables that might influence the observed values). To sum up our discussion, RMSE is a good measure to use if we want to estimate the standard deviation σ of a typical observed value from our model’s prediction, assuming that our observed data can be decomposed as: To phrase it another way, RMSE is a good way to answer the question: “How far off should we expect our model to be on its next prediction?” By dividing by n, we keep this measure of error consistent as we move from a small collection of observations to a larger collection (it just becomes more accurate as we increase the number of observations). 
We should also now have an explanation for the division by n under the square root in RMSE: it allows us to estimate the standard deviation σ of the error for a typical single observation rather than some kind of “total error”. But then RMSE is a good estimator for the standard deviation σ of the distribution of our errors! This tells us that Σᵢ (ŷᵢ - yᵢ)² / n is a good estimator for E = σ². In fact a sharper form of the central limit theorem tell us its variance should converge to 0 asymptotically like 1/n. The central limit theorem tells us that as n gets larger, the variance of the quantity Σᵢ (ŷᵢ - yᵢ)² / n = Σᵢ (εᵢ)² / n should converge to zero. Notice the left hand side looks familiar! If we removed the expectation E from inside the square root, it is exactly our formula for RMSE form before.
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