The sample variance is, in turn, is often composed of two main components

s²_sample = s²_object + s²_sampling

The object variance is an indication of heterogeneity while the sampling variance is a result only of the process of taking grabs from the object.

To determine heterogeneity, one must insure that the measurement process has

s²_measurement << s²_object

and that sample grab size is sufficient for

On the other hand, for the analysis precision to be limited by the measurement precision (e.g., that due to the instrument or method) for homogeneous objects

s²_sampling << s²_measurement

To insure that sampling does not result in less precision than is implicit in the method requires a little theory. This theory can help use determine the size of grab samples required for a given level of precision.

Sampling variance arises due to a statistical fluctuation in the number of "units" of atoms or particles that contain the analyte.

This fluctuation is rigorously expressed by the binomial probability distribution

p is the probability (out of one) that a target unit (atom, particle, etc.) is obtained in a given random grab sampling. k is the total number of target units in a sample grab of n units. For example, a coin has a p=1/2 out of 1 chance of coming up heads; a 6-sided die has a chance of p=1/6 put of 1 chance of showing the number "4". Similarly, a 1:1 mixture of stainless and mild steel ball bearings has p=1/2 chance of sampling a stainless bearing; a 1:5 mixture has a stainless probability of p=1/6.

For large grab sizes, where n is very large, the probability of obtaining k target units out of a total of n units is approximated by the Gaussian approximate of the binomial probability distribution

By comparison to the Normal Distribution, one may see that the mean and variance are related to the probability and number of units by