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I'm trying to compare the variability within two samples which are quite different in scale (pupil capacity of different types of educational establishment (nursery, primary and secondary schools). My approach is to calculate the standard deviation for each sample, then divide that by the sample mean.

I've not seen this metric used, so I'm wondering is there an accepted name for it? Or is there is a better metric to use which would explain why I haven't seen it?

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Standard deviation divided by mean is called coefficient of variation. It is defined exactly as you did

$$ c_{\rm v} = \frac{\sigma}{\mu} $$

in terms of population mean and standard deviation, or it can be estimated using sample mean and sample standard deviation.

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    $\begingroup$ And this measure has always been problematic because it is highly sensitive to the origin for the variable. An extreme example would be temperature measurements in degrees C vs. F. $\endgroup$ – Frank Harrell Oct 11 '16 at 12:52
  • $\begingroup$ @FrankHarrell what about calculating the mean of each sample, and then subtracting this mean from each value in the (corresponding) sample - to set each sample at mean = 0 - and then recalculating the standard deviation on these translated datasets? $\endgroup$ – Mooks Oct 11 '16 at 14:53
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    $\begingroup$ @Mooks ...and then dividing by zero? $\endgroup$ – Tim Oct 11 '16 at 14:56
  • $\begingroup$ No I'm talking about only calculating the sd and comparing those, because then the different scales aren't a problem - I'm not sure if this is a way around the problems with the Cv mentioned above (or whether it's subject to the same issues). $\endgroup$ – Mooks Oct 11 '16 at 15:23
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    $\begingroup$ Frank's criticism seems to be another way of saying "don't use this with measurements that have an arbitrary origin" ... a sentiment it's hard to disagree with. The many cases where it makes sense don't have that. $\endgroup$ – Glen_b Oct 12 '16 at 0:28

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