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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?

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Let $X$ be a random variable, with mean $\mu$ and variance $\sigma^2$.

By the properties of the variance operator, the variance is unchanged.

$$\operatorname{Var}(X-\mu) = \operatorname{Var}(X) - \operatorname{Var}(\mu) - 2\operatorname{Cov}(X,\mu) = \operatorname{Var}(X) - 0 - 0$$

This means the error bars remain unchanged.

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  • $\begingroup$ Ok, I see your point. Could you help me see whats wrong about my argument i.e. let y_i = c + e_i with e_i ~ N(0, s_i2). Estimation of c delivers c_est with an standard error of s_c. Subtracting the c_est of y yields the residuals r_i = y_i - c_est. Now I estimate the errorbar of r_i with gaussian errorpropagation i.e. s_r_i = sqrt(s_i2 + s_c**2). $\endgroup$ – numb_papaya Jun 5 at 14:27

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