Calculating ensemble value for data with non-gaussian errors

I have 10 data points each of which have non-symmetric (i.e., non-Gaussian) errors. That is, if 'a' is the central value (representing the median), then a^{+b}_{-c} means that 'b' is the positive 1-sigma error and 'c' is the negative 1-sigma error. I want to be able to get a representative value with positive and negative errors for the ensemble of 10 points. How does one do this?

The reason I ask is because I have some constant number which I want to compare to the representative value and uncertainty of the ensemble.
I have ten data points with non-gaussian errors. I want to somehow combine these ten data into one datum that would be representative of the total ten. Given the non-symmetric nature of the error bars, how can one do this?


Ambassador to the humans
I guess that depends on how you measure if the value is representative. Typically for non symmetric data we would just use the median.
Thank you for your response.

So currently all the ten data have a median value and 1-sigma errors which are asymmetric. What I want is a robust statistical representation of the ten data in the form of one data point. I can imagine that the median value for the representative point could just be the mean of the 10 median values of the data. But how do I then get the asymmetric errors on this point?
Basically, I have a posterior probability distribution for some quantity. The central value of the data is the median value of the posterior distribution, and the 1-sigma values are computed numerically and represent ~34% (in analogy to the gaussian 1-sigma) of the solutions to either side of the median value. But because the posterior distribution could be skewed and is not generally gaussian, the error bars are non-symmetric.


TS Contributor
how about combining the distributions using the total probability formula and calculating the mean / median of the resulting distribution?


TS Contributor
Thanks for the response. Can you be more specific?
if you have 10 distributions f1(x)...f10(x) and you have an estimate of the relative frequencies of when each distribution applies p1 ..p10 then you could have f(x)=SUM(pi*fi(x)) where SUM(pi)=1. Then use f(x) to calculate the mean of x or the median of x or, even better, the mode.

Thank you for your nice response. I would go about doing it but then there is ambiguity in assigning p1, p2, ..., pn. I wonder if you might know how to use a chi-square test on the ten data with their asymmetric errors to test if their total representative value is consistent or not with some constant reference value (that doesn't have errors, but is just a number).