# How do the errorbars change when time series is mean subtracted?

I want to subtract the mean from my time series. Each data point has a corresponding errorbar. I calculate the mean by fitting a constant with a MLE estimation and estimate the standard error with the inverse of the fisher matrix. If I subtract the mean, do the errorbars of the residuals change? If I use gaussian error propagation then the errorbar of each data point needs to be quadratically summed with the standard error. Is this the correct way?

Let $$X$$ be a random variable, with mean $$\mu$$ and variance $$\sigma^2$$.
$$\operatorname{Var}(X-\mu) = \operatorname{Var}(X) - \operatorname{Var}(\mu) - 2\operatorname{Cov}(X,\mu) = \operatorname{Var}(X) - 0 - 0$$