# Errorbar determination for non gaussian distribution

Suppose I have a data set which has a nice Gaussian distribution f(x), then I can "summarize" as

Mean{f(x)} +- Std{f(x)}

Where std stands for standard deviation. However, If my data does not look like a nice Gaussian distribution the concept of standard deviation breaks down.

Suppose I have a data distribution which looks like this:

Where the black dotted line represents the mean of the data. What would, in this case, be the "best" choice for my error bars if I want to represent the data in a similar fashion as with the Gaussian distribution and why?