# Gamma distribution and 68–95–99.7 confidence limits

What is the equivalent of $\sigma$ confidence limits for a normal distribution (68–95–99.7 %) on a gamma distribution ?

And is there a way relate it to the standard deviation and mean (or median) of the distribution, or we have to make a fit on the distribution (since I suppose errorbars will be asymetric) ?

Thank you very much in advance for your help.

• stats.stackexchange.com/questions/89230/… Sep 22, 2017 at 14:33
• Since the gamma is asymmetric it's not quite clear what you intend by "equivalent". What is it you need this "equivalent" to tell you? Sep 22, 2017 at 15:34
• You seem to be confusing a rule of thumb about probability distributions with some kind of statement about confidence limits. Could you clarify what you're referring to? If you really are referring to the standard 68-95-99.7 rule (which holds for much more than Normal distributions, btw), then could you specify which Gamma distributions you are concerned about? This rule applies nicely to Gamma distributions with shape parameters greater than 20 but not so well to the other Gamma distributions, especially with parameters less than 2.
– whuber
Sep 22, 2017 at 17:39
• Well, approximately holds under some suitable conditions. I don't mind using the 95 rule under unimodality (IMO it performs somewhat better than the other two), but even then it's sometimes a bit rough; IIRC the bounds are between 88.9 and 100% Sep 23, 2017 at 1:24

• Could you clarify why you expect the rule to change with the scale? Because it expresses differences as multiples of $\sigma$ (presumably the SD), it is explicitly scale-independent. Are you perhaps implicitly remarking on the confusion in the question between confidence limits and standard deviation limits?