I have a very basic question about standard deviations.

I have information on different units, $i$ over time horizon $t$ (I have panel data). Let me denote the variable of interest as $y_{it}$ which I observe for each individual. I want to see whether or not there is a relationship between the average value of $y_i$ across individuals and the standard deviation of $y_i$ . In other words, I want to calculate the mean and standard deviation of this variable for each individual and the see if there is a relationship between the mean and variability of this variable. I graphed these variables and I was surprised to see a very clear positive relationship between the mean and standard deviation. My question is: is it surprising? Did I make a conceptual mistake? I mean individuals with a higher mean value of $y_i$ will have a higher variance just because of the fact that their values of $y$ in general are bigger? In other words, should I standardize the mean as well? Thanks!

  • $\begingroup$ Just as a side not, I just plotted the standard deviation of y on the x-axis and the standardized mean on the y-axis and I get a decreasing relationship $\endgroup$ – ChinG Nov 4 '15 at 15:41
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    $\begingroup$ Standard deviations are not scale invariant. One measure of variability that is scale invariant is the coefficient of variation, CV, defined as the ratio of the std dev to the mean. CVs of differing metrics of directly comparable whereas std devs are not. $\endgroup$ – DJohnson Nov 4 '15 at 15:45
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    $\begingroup$ Echoing @DJohnson's helpful comment, SD proportional to mean implies constant CV. That in turn often implies use of logarithmic scale. For more, see e.g. stats.stackexchange.com/questions/118497/… $\endgroup$ – Nick Cox Nov 4 '15 at 15:50
  • $\begingroup$ So with regards to my question, would it be reasonable to plot the CV against the mean? $\endgroup$ – ChinG Nov 4 '15 at 15:53
  • $\begingroup$ You could do that, but it's just the same information. The implication as said is to consider logarithmic scale, which could imply transformation or it could mean a model with logarithmic link function. Transformation requires values to be all positive. $\endgroup$ – Nick Cox Nov 4 '15 at 17:21

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