# How to quantify the fluctuation of an error?

First of all sorry for the bad title, but unfortunately, I can't think of a better one at the moment. Hopefully, that will change when my question is answered.

Let's say I have two sets of values: The target values $$T_i$$ and the actual values $$X_i$$ and I want to quantify the errors $$E_i=X_i-T_i$$ using a single number. Due to the fact that I don't want negative and positive to compensate each other and the fact that I'd like to punish outliers, I decided to use the Root Mean Square Error $$\text{RMSE}$$.

Now the question: How can I quantify the fluctuation of the errors in a way that can be compared with the $$\text{RMSE}$$? My first idea was to use the standard deviation. After all, the $$\text{RMSE}$$ is a mean value and it is a plausible assumption, that the errors $$E_i$$ are normally distributed. However, I think the standard deviation is "incompatible" with the $$\text{RMSE}$$, because unlike the $$\text{RMSE}$$ it does not operate on the squared errors.

EDIT: I'll try to make things clearer. What I want to do is to quantify the error of the $$\text{RMSE}$$. After all it is a mean, so there must be a way to tell how much the values, this mean is based on, fluctuate. Let's say I would use the $$\text{MAE}$$ instead of the $$\text{RMSE}$$. Then I could easily quantify the fluctuation of the absolute errors using their standard deviation. However, to apply the same approach to the $$\text{RMSE}$$, I would have to calculate the square root of the standard deviation of the squared errors, which seems a little odd to me.