Variance

Gy defines the fundamental variance for a sample S to be the rela­tive vari­ance of the sample’s critical con­tent, a_S, when the sample is correct and frag­ments are selected for S inde­pend­ently of each other (i.e., in inde­pend­ent Bernoulli trials). He esti­mates the funda­mental vari­ance using the follow­ing equation.

The 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 “uncertainty propagation” or “error propa­ga­tion.” Using error propa­ga­tion, one esti­mates the rela­tive vari­ance of a_S by

When one makes Gy’s assump­tions, with the selec­tion prob­abil­ity for each frag­ment equal to p = m_S / m_L, and sub­sti­tutes exact expres­sions for Var(A_S), E(A_S), Var(m_S), E(m_S), and Cov(A_S, m_S) into Equation 2, one obtains Equation 1 for the funda­mental variance.

One may also derive an expres­sion for the sam­pling vari­ance under the assump­tion that S is a random sample of exactly k frag­ments from the lot (for some positive number k). The fact that S is a random sam­ple of size k means that for all sub­sets G of L of size k,

Under 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, S, seems also to work well enough for a fair sample, S_F, of size k such that for all sub­sets G of L of size k,

And of course, S_F has the advantage of being provably unbiased, so that 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.

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 N / (N − 1), which is near unity when N is large, and there­fore can be neglected.

Generally, one may expect Equation 1 to be a good approxi­ma­tion as long as the rela­tive stand­ard devia­tion of m_S is small. If RSD(m_S) is large, say because the sample is too small or there are some very massive frag­ments in the lot, the approxi­ma­tion may be much worse.

It is proved elsewhere that when the sam­pling is cor­rect, keep­ing RSD(m_S) small also ensures that the sampling bias is negli­gible in com­parison to the stand­ard deviation.

* Well, there is the minor issue of the empty sample. If one selects 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.