In unbiased data, RMSE and standard deviation is same?
Let $x_{i}$ be your true value for the $i^{th}$ data point and $\hat{x}_{i}$ the estimated value. If we assume that the differences between the estimated and true values have
mean zero (i.e. the $\hat{x}_{i}$ are distributed around $x_{i}$) (Unbiasd)
follow a Normal distribution
and all have the same standard deviation $\sigma$
in short:
$$\hat{x}_{i}-x_{i} \sim \mathcal{N}\left(0,\sigma^{2}\right),$$
Then, RMSE between $x_{i}$ and $\hat{x}_{i}$ is $$ \text{RMSE}^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i-\hat{x}_i)^2\,, $$
Standard deviation of $x_{i}$ - $\hat{x}_{i}$ is $$ \text{SD}^2 = \frac{1}{n}\sum_{i=1}^{n}((x_i-\hat{x}_i)-mean)^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i-\hat{x}_i)^2\ $$
So i think RMSE and SD is same in unbiased data. However, i can not be sure and find any reference.
