# Errorbars on a weighted average (of a non gaussian distribution)

I have some datapoints $x_i$ and errors $\sigma_i$ which are the absolute value of some datapoints $w_i$ with (supposed) gaussian errorbars (the $\sigma_i$). The $x_i$ are amplitudes so we did not care the sign.

I need to compute the average (and weighted by $1/\sigma_i^2$ average) of them. And I need the errorbars on this mean.

How can I compute it if I don't have the values $w_i$ but just $x_i$?

I mean if this was just the error of gaussian distributions (i mean for the $w_i$ ) I would use the weighted mean for this formula:

$$\mu = \frac{\Sigma w_i \sigma_i^{-2}}{\Sigma \sigma_i^{-2}}$$

And for the error $$\sigma = \sqrt{\frac{1}{\Sigma \sigma_i^{-2}}}$$ (as in https://ned.ipac.caltech.edu/level5/Leo/Stats4_5.html)

But what about in this case with the $x_i$?

• Welcome to the site! You can analytically derive the density of x $P(X<\alpha) \equiv P(-\alpha < w < \alpha) = F_w(\alpha) - F_w(-\alpha)$. Alternatively, you can bootstrap; sample $x_i$ and estimate the variance of the weighted mean. – Emre Sep 13 '17 at 19:16