Ratio of standard deviation to mean

Question

Let's say that we have a confidence interval like this: µ ± σ, where σ is the standard deviation and µ is the mean. I want to know if the ratio σ/µ has any interpretation? Does it mean anything if we evaluate it? Thank you in advance!

I've added the "coefficient of variation" tag to your post. It may cover some properties that you are interested in. — Sycorax, Jul 12 '15 at 02:14
Pay attention to the usage of notations. $\mu$ and $\sigma$ are usually used as parameters, which should not appear in confidence intervals. — Zhanxiong, Jul 12 '15 at 05:16

small_data88 · Accepted Answer · 2015-07-12T19:20:04.270

6

Yes, it does have an interpretation. That ratio is called the Coefficient of Variation(CV) and it shows the magnitude of variation in relation to the population mean. It is used to describe the variation within the data without depending on the measurement unit of the data, so you can easily compare the dispersion across different distributions of data Please check out CV wikipedia or This Link for more information. NOTE: This statistic is only meaningful for ratio variables, which are variables that have a clear definition of 0(e.g. 0 means none of that variable).

edited Jul 12 '15 at 19:20

answered Jul 12 '15 at 01:48

small_data88

226
1
6

2

But note that the CV is only a meaningful number when computed for ratio variables. These are variables like length or weight or enzyme activity, where a value of 0.0 means there is none of that variable. If 0.0 is basically arbitrary, as it is in temperature in C or F, then the CV would be meaningless. – Harvey Motulsky Jul 12 '15 at 14:01
@Harvey Motulsky yea I saw that, which is why I pointed the OP to check out the wikipedia page or the other link for further information :) – small_data88 Jul 12 '15 at 15:02

score 5 · Answer 2 · answered Jul 12 '15 at 11:57

The inverse of this generalizes to n dimensions https://en.wikipedia.org/wiki/Mahalanobis_distance , and basically is a distance scaled by the covariance matrix. It tells you how many "sigmas" out a point is ellipsoidally along its covariance. In this case, "sigma" means sqrt(eigenvalue). Square root of eigenvalues are the standard deviations in the "independent" coordinate frame, i.e., when the covariance is rotated so that its eigenvectors align with the coordinate axes.

Ratio of standard deviation to mean

2 Answers2