# RMSE punishes large errors

Every text book will usually say that RMSE punishes large errors (because of the quadratic nature of it).

What if the average errors are all between 0 and 1? Example: we measure temperature and we get very good predictions. Our MAE is around 0.5 - in that case RMSE punishes low errors. Am I missing something out?

• Why do you say that it 'punishes low errors' ? 0.01 squared is 0.0001 and 0.99 squared is 0.9801, the latter is a many times larger than the former ?
– user83346
Jan 22, 2016 at 9:24
• To put @fcop's comment another way, if most of your errors are on the order of $1/100$, an error of $99/100$ is still large by comparison. Jan 22, 2016 at 13:21

## 1 Answer

RMSE will punish large errors, even if the domain is restricted from 0 to 1. To test this, try computing the squared error for a couple of different cases.

If error is 1, squared error is 1. If error is 0.5, squared error is 0.25. If error is 0.1, squared error is 0.01.

As you can see, the squared error for an error of 1 is 100 times the squared error for an error of 0.1.

• +1. It is the same quadratic whatever the range of errors. The curve gets steeper and steeper away from 0. Therefore RMSE punishes large errors, always. Jan 22, 2016 at 13:27