rmsstandard deviation
RMSE (Root mean square error) and SD (Standard deviation) have similar formulas.
This link says
The only difference is that you divide by $n$ and not $n−1$ since
you are not subtracting the sample mean here. The RMSE would then
correspond to $\sigma$ . Therefore, the population RMSE is $\sigma$
and you want a CI for that.
So I want to know whether RMSE and SD are the same. Also, I want reference about it.
Let's say that our responses are $y_1, \dots, y_n$ and our predicted values are $\hat y_1, \dots, \hat y_n$.
The sample variance (using $n$ rather than $n-1$ for simplicity) is $\frac{1}{n} \sum_{i=1}^n (y_i - \bar y)^2$ while the MSE is $\frac{1}{n} \sum_{i=1}^n (y_i - \hat y_i)^2$. Thus the sample variance gives how much the responses vary around the mean while the MSE gives how much the responses vary around our predictions. If we think of the overall mean $\bar y$ as being the simplest predictor that we'd ever consider, then by comparing the MSE to the sample variance of the responses we can see how much more variation we've explained with our model. This is exactly what the $R^2$ value does in linear regression.
Consider the following picture:
The sample variance of the $y_i$ is the variability around the horizontal line. If we project all of the data onto the $Y$ axis we can see this. The MSE is the mean squared distance to the regression line, i.e. the variability around the regression line (i.e. the $\hat y_i$). So the variability measured by the sample variance is the averaged squared distance to the horizontal line, which we can see is substantially more than the average squared distance to the regression line.
RMSE is the square root of MSE. But the answer to your question depends on if you are talking about the MSE of a predictor versus an estimator.
MSE of an estimator is a fixed quantity, and has no variance. So it makes no sense to talk about the SD of the MSE.
Consider a special case with model, $M_1$, $Y_i = \beta$ for $n$ $iid$ observations. Suppose $E[Y_i]=\beta$ and $Var[Y_i] = \phi$, $\forall i$. An unbiased estimator is for $\beta$ is $\hat \beta = n^{-1} \sum_i Y_i$. Now $\hat \beta $ is a function of a random sample and so is random itself. If it's random, it has variance. Since it is unbiased, $MSE[\hat \beta]=Var[\hat\beta] = \phi$. So the MSE is constant.
$Var\big[MSE[\hat\beta]\big]=Var\big[\phi\big]=0$.
This is a function of a random sample so it is itself random and therefore has variance. Consider predictor from the model above, $M_1$
$MSE_{pred} = n^{-1} \sum_i(Y_i - \hat Y)^2 = n^{-1} \sum_i(Y_i - \hat \beta)^2 $
$\hat \beta$ is a function of data (random). $Y_i$ is the data (also random), so the whole MSE is a statistic - so is itself random. So it has variance and we can meaningfully talk about the SD of the MSE.
Best Answer
TLDR; While the formulas may be similar, RMSE and standard deviation have different usage.
You are right that both standard deviation and RMSE are similar because they are square roots of squared differences between some values. Nonetheless, they are not the same. Standard deviation is used to measure the spread of data around the mean, while RMSE is used to measure distance between some values and prediction for those values. RMSE is generally used to measure the error of prediction, i.e. how much the predictions you made differ from the predicted data. If you use mean as your prediction for all the cases, then RMSE and SD will be exactly the same.
As a sidenote, you may notice that mean is a value that minimizes the squared distance to all the values in the sample. This is the reason why we use standard deviation along with it -- they are related species!