Did a Turbidity - suspended sediment concentration (SSC) calibration. Did 8 replicates; tried to find out the relationship between coefficient of variation (CV) and number of replicates. For each number of replicates (eg. 2/8), 2 random samples were chosen out of 8. This was bootstrapped 10000 times and an average was obtained.
Results show that as replicates increases, CV increases, which is opposite to initial expectation.
How to interpret this?
What is the cause of this?
Is there a way to determine the optimum number of samples?
This is an interesting question.
As the coefficient of variation (CoV) depends on the first two moments, the mean and standard deviation, how might each be affected by the number of replicates?
Sample CV underestimates parameter CV, particularly when the sample size is small. If this cannot explain what you observed, I would check the bootstrap algorithm that may have a limitation when the number of observations is small.
With increasing repeatability, the number of repetitions increases, but the interval (range) of changes remains unobstructed.
Considering the minimum-variance unbiased estimator of (μ/σ)^2 with normally distributed data, Johnson, Kotz, and Balakrishnan, Continuous Univariate Distributions, Vol. 2 (1995), following Rukhin (1992), indicate that the estimate s/xbar (where s^2 is the unbiased estimate, dividing by (1 — N)) increases with N asymptotically as
s/xbar ~ σ/μ [1 —1/Ν —0.5 1/Ν (σ/μ)^2].
If this doesn’t explain the increase you have found, it might not be too cumbersome to replicate your analysis with normally generated data so as to check your algorithms, bootstrap procedure, and normality assumption. You may find that data sets exhibiting the increase you found are not uncommon.
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