## VarianceGy defines the fundamental variance for a sample a_{S}, when the
sample is correct and fragments are selected for independently of each
other (i.e., in independent Bernoulli trials). He estimates the fundamental
variance using the following equation.SThe fundamental variance is considered to be the smallest relative sampling variance that is practically achievable without increasing the sample size or reducing the fragment sizes (i.e., grinding or milling the material before sampling). In routine practice one can expect the sampling variance to be somewhat larger than the fundamental variance, but any additional variance tends to be harder to estimate. Equation 1 can be derived by a technique sometimes called When one makes Gy’s assumptions, with the selection probability for each
fragment equal to var(,
A)_{S},
E(A)_{S}var(,
m)_{S}, and
E(m)_{S}cov( into Equation 2, one obtains Equation 1 for the fundamental variance.A, _{S}m)_{S}One may also derive an expression for the sampling variance under the
assumption that fragments
from the lot (for some positive number k).
The fact that k is a random sample of size S means that for all
subsets k of G of size L,kUnder this assumption Equation 2 yields a slightly different expression for the relative variance, which is shown below. Equation 4, which was derived for a correct sample, , of size S_{F} such that for all
subsets k of G of size L,kAnd of course, .E(a_{SF}) = a_{L}Both Equation 1 and Equation 4 are only approximately true, but they can be applied to many real-life situations in the laboratory. There is another equation that is exactly true, but which obviously does not apply to many real-life situations in the lab.
Theorem Suppose
and m_{1} = m_{2} =
⋯ = m_{N} is a sample such
that SPr[ whenever
S = G] = Pr[S = H]|.
Then
G| = |H|When all the fragment masses are equal, selecting fragments for the sample
in independent Bernoulli trials makes the premise of the theorem true.*
So, in this case at least, Gy’s equation for the
fundamental variance is a good approximation, except for the missing factor
is large, and therefore can be neglected.NGenerally, one may expect Equation 1 to be a good approximation as long as the relative
standard deviation of It is proved elsewhere that when the sampling is correct, keeping
* Well, there is the minor issue of the empty sample. If one selects all the fragments in independent Bernoulli trials, it is possible that no fragments will be selected at all. In that case the sampling must be repeated until a nonempty set of fragments is obtained, but the premise of the theorem is still true. It is possible to avoid the problem by selecting the first fragment “correctly” and then selecting all the other fragments in independent Bernoulli trials using Gy’s procedure. The sampling is still “correct” and there is no danger of an empty sample. Or, to ensure the sample is truly unbiased, select the first fragment with each fragment’s selection probability proportional to its mass. This approach theoretically produces a nonempty and unbiased sample, but it is even more impractical than the original. | |||||||