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I have a distribution in which the standard deviation is positively correlated with the mean (0.9). The stdev increases as the mean increases. The data is count data without zeros. I'm interpreting this as a interesting fact, but I want to make sure I am not missing something here.

How rare is this? Is it normal to find somewhat of a correlation between stdev and mean? Are there specific distributions that have this correlation?

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    $\begingroup$ You answered one particular case in your question: The data are counts. Counts are non-negative and therefore usually right-skewed. For example, in the Poisson distribution the mean is the variance. $\endgroup$
    – Frans Rodenburg
    Commented Sep 6, 2019 at 9:00
  • $\begingroup$ @FransRodenburg Thanks for your insights. Can I in that case deduce anything interesting from the standard deviation? Or is there a better alternative to measure the variability of the counts? $\endgroup$
    – don.joey
    Commented Sep 6, 2019 at 9:24
  • $\begingroup$ Finding an alternative really depends on what you are trying to achieve. This is a sufficiently different question that I suggest posting a new question with more detail about your problem. $\endgroup$
    – mkt
    Commented Sep 6, 2019 at 9:33
  • $\begingroup$ Unless I'm being dense, one distribution has one mean and one standard deviation. There's no chance for either to change or to be correlated within that distribution. Maybe you have multiple distributions. Could you display your data, or a sample of them? $\endgroup$
    – rolando2
    Commented Sep 6, 2019 at 12:46
  • $\begingroup$ It is rare for the SD and mean not to be dependent. $\endgroup$
    – whuber ♦
    Commented Sep 6, 2019 at 14:32

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It's fairly common with count data, or more generally data that are bounded at zero. Frans Rosenburg points out the best example, the Poisson distribution, where the mean and variance are identical.

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  • $\begingroup$ Thanks, does that mean the standard deviation is not insightful with Poisson distributions? $\endgroup$
    – don.joey
    Commented Sep 6, 2019 at 9:24
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    $\begingroup$ @don.joey Yes, in the sense that it does not provide you any new information if you already know the mean. Note that the assumption of identical mean & variance in the Poisson is quite strict, and not all count data obeys that. So a common approach to modelling count data is to use a distribution that relaxes this assumption a bit. To quote this answer by jbowman: "For a quasi-poisson regression, the variance is assumed to be a linear function of the mean; for negative binomial regression, a quadratic function." There are other options too. $\endgroup$
    – mkt
    Commented Sep 6, 2019 at 9:30
  • $\begingroup$ For the poisson the theoretical mean and variance are identical, not necessarily the empirical ones --- they should be correlated! $\endgroup$
    – kjetil b halvorsen ♦
    Commented Nov 23, 2019 at 11:40

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