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I have a number of data points (which can be small) with measurement error bars. I assume the intrinsic values are drawn from some sort of normal distribution.

What is the most robust method for calculating the width of this distribution (i.e. the dispersion) and its uncertainty, removing the additional spread from the error bars?

Including the additional dispersion in a $\chi^2$ minimisation (or MCMC) doesn't appear to work as the dispersion just goes to infinity to reduce the $\chi^2$. Perhaps there is some approach where I could examine the cumulative probability for a $\chi^2$ distribution against the measured $\chi^2$, increasing the error bars, but I'm not convinced this is valid.

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I think I managed to work this one out. I was assuming that the log likelihood of my dispersion was simply $-\chi^2/2$. If the size of the data $\sigma$ changes, it goes to $-\chi^2/2 + \sum \log(1/\sqrt{2 \pi \sigma^2})$. If this likelihood is used in an MCMC analysis, it seems to produce reasonable results (though I need to check further).

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