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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:

enter image description here

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?

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The standard deviation exists for (almost) all distributions, and a sample standard deviation can be calculated for every sample.

What you should visualize depends on what you want to visualize. You could visualize the mean and a measure of concentratedness of your data. In such a case, I would plot a dot for the mean and a line covering a central quantile, e.g., extending from the 5% to the 95% quantile of your data. If you have enough space, you could give a boxplot, potentially with an additional dot to indicate the mean. Or a beanplot to give an indication of the entire distribution.

Alternatively, you might want to indicate how well defined your mean is, which is something different although related to the above. In this case, you could plot the mean and a line giving a confidence interval for the mean. If you assume your mean is normally distributed, then this line extends by a scaling factor times the standard error of the mean, which in turn is a function of the standard deviation of your data and the sample size. However, your confidence interval could also be asymmetric, e.g., if you got it from a bootstrap.

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  • $\begingroup$ Suppose I have multiple datasets very similar to the one shown above. What I would like to do in the end, is to plot the mean of each of these datasets with a certain errorbar in a scatter/line plot. Thus indeed, I would like to indicate in this manner how well defined the mean is. Is it then statistically "allowed" to use the standard error of the mean as an error bar? Or should the standard deviation itself be used? $\endgroup$ – Guido De Haan Aug 8 '18 at 8:20
  • $\begingroup$ In a case like that, I would plot the means and associate a line of length plus/minus a certain multiple of the SEM to show a confidence interval for each mean, with a coverage you can specify. Unless your distribution is rather pathological or your sample size small, this should be valid by the CLT. $\endgroup$ – Stephan Kolassa Aug 8 '18 at 12:41

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