# What does a squared coefficient of variation signify?

The coefficient of variation is defined as the ratio of S.D to mean, a dimensionless number which on multiplication by 100 gives the percentage of dispersion with mean. But when we square this coefficient of variation, what does it signify?

(I have to understand a graph having squared coefficient of variation on y axis and mean on x axis for Poisson distributed data).

Thanks

-
The coefficient of variation is $\frac{\text{sd}(X)}{\text{mean}(X)}$. This is the same as $\text{sd}(\frac{X}{\text{mean}(X)})$. If you square this you just get the variance of $\frac{X}{\text{mean}(X)}$, i.e. X expressed in units of $\text{mean}(X)$. Does this help you? –  Erik Apr 24 '12 at 7:17
Thankyou Erik for your quick response... –  bioinformatician Apr 24 '12 at 8:24
I am just a beginner... I want to confirm my basic understanding (1) suppose i have four numbers i.e 5,7,9,11 (2) mean of these numbers comes out to be 8 (3) variance calculated as (8-5)^2 + (8-7)^2 + (8-9)^2 + (9-11)^ / 4 = 5 and the S.D is underroot of 5 i.e 2.23 (5) the coefficient of variation : 2.23/mean(=8) will be .2795 (6) if i square .2795 , i get ~ .078 ... so how to comprehend this .078 in terms of my data (5,7,9,11) ? –  bioinformatician Apr 24 '12 at 8:34
Are you sure this is a squared CV and not the ratio of the sample variance to the sample mean? For a Poisson distribution, this ratio should be close to 1, so it could make sense to plot it against the mean as a check for departures from "Poissonness": the plot should be approximately level. Plotting the squared CV would instead produce a hyperbolic curve, which is not a good visual reference for exploring and understanding data. –  whuber Apr 24 '12 at 13:12