Analysis and Sample Variances

The total or analysis variance is generally composed of two main parts

s2analysis = s2sample + s2measurement

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

s2sample = s2object + s2sampling

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

s2measurement << s2object

and that sample grab size is sufficient for

s2sampling << s2object

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

s2sampling << s2measurement

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

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

msampling = np
s2sampling = np(1-p)

Both measurement mean, m, and variance, s2, increase with sample or grab size. On the other hand, the relative standard deviation (RSD) due to sampling is

Here, the RSD improves, i.e., gets smaller, with increased sample grab size. The RSD varies also with probability, p. For unit probability (p=1), the RSD is zero. There is no sampling variance if the object is homogeneous. For small probability, that is for trace analysis, the RSD is

the RSD is inversely proportional to the square root of the number of target or anayte particles. In this case, increasing grab size will increase relative precision.

A Priori Determination of Sample Grab Size

To insure that the measurement precision is not due to sampling (after all, who cares about statistical fluctuations due to sampling?), it is usually sufficient to have the relative sampling precision be less than 10% of that of the measurement. Mathematically,

RSDsampling £ (0.1) RSDmeasurement

Substitution of sampling RSD,

Or, in terms of the measurement standard deviation

This page was edited Tuesday, August 03, 2004