I am analyzing an engineering problem where the the coefficient of variation (CV) is being used to assess how reliably the quantity of a certain chemical can be measured. The context of this problem is as follows. The engineers have 16 stations and at each station they take about 100 samples. They measure the amount of a specific chemical in the samples. For each of the 16 sites, the sample mean and sdev of the amount of chemical in the samples are computed.
The engineers think there is an inherent problem with the how reliable the measurements are because the CV is very high. I believe the solution to their problem is simply that they are looking at a Poisson process. The basis for this belief is that the chemical measured is produced by scale in a pipe that "flakes off" occasionally (hence the measured quantity can be seen as the sum of a number of discrete events and the conditions for a Poisson process are most likely satisfied).
This hypothesis is testable since for a Poisson distribution the sdev is equal to the square root of the mean. When I make a scatterplot of the sdev vs the sqrt of the mean (for the 16 stations), there is a strong positive linear relation (with a coeff of det about 73% and a p-value for the model about 0.00002)
My problem however is the following: when I look at the estimated slope for the regression model (sdev vs sqrt(Xbar)) it is about 11, although theory predicts it should be unity. I am wondering what would account for this? I do not know of another distribution that has the property that the sdev and the square root of the mean are proportional and I don't think a non-homogeneous Poisson process would produce my observed results.
Addition: when I try to fit the sdev and the mean with a linear model (to see if an exponential model might be appropriate) the fit is also very good (R^2 is about 70% and pvalue for the model is about 0.00005) but the slope again is off but not by as much (is 2.3, but should be unity). Which is at least better and perhaps the discrepancy can be explained by random error (the standard error is about 0.4, so 1 is close to being with a 95% CI for the slope). I'm not sure the model makes sense physically though.
Thanks for any help or suggestions anyone might have,