8/6/2019 Analysis and Sample Variances

1/4

Analysis andSample Variances

The total or analysis variance is generally composed of two

main parts

2analysis = 2sample +

2measurement

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

main components

2sample = 2object+

2sampling

The object variance is an indication of heterogeneity whilethesamplingvariance is a result only of the process oftaking grabs from the object.

To determine heterogeneity, one must insure that themeasurement process has

2measurement

8/6/2019 Analysis and Sample Variances

2/4

theory can help use determine the size of grab samples

required for a given level of precision.

SamplingVariance

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.

kis the total number of target units in a sample grab ofn

units. For example, a coin has ap=1/2 out of 1 chance ofcoming up heads; a 6-sided die has a chance ofp=1/6 putof 1 chance of showing the number "4". Similarly, a 1:1

mixture of stainless and mild steel ball bearings hasp=1/2

chance of sampling a stainless bearing; a 1:5 mixture has a

stainless probability ofp=1/6.

For large grab sizes, where n is very large, the probability

of obtaining ktarget units out of a total ofn units isapproximated by the Gaussian approximate of the binomial

probability distribution

8/6/2019 Analysis and Sample Variances

3/4

By comparison to theNormal Distribution, one may seethat the mean and variance are related to the probability and

number of units by

sampling= np

2sampling= np(1-p)

Both measurement mean, , and variance, 2, increase

with sample or grab size. On the other hand, the relativestandard deviation (RSD) due to sampling is

Here, theRSD improves, i.e., gets smaller, with increased

sample grab size. TheRSD varies also with probability,p.For unit probability (p=1), theRSD is zero. There is nosampling variance if the object is homogeneous. For small

probability, that is for trace analysis, theRSD is

the RSD is inversely proportional to the square root of thenumber of target or anayte particles. In this case, increasing

grab size will increase relative precision.

http://www.chem.usu.edu/~sbialkow/Classes/3600/Overheads/Normal/Distribution.htmlhttp://www.chem.usu.edu/~sbialkow/Classes/3600/Overheads/Normal/Distribution.html

8/6/2019 Analysis and Sample Variances

4/4

A PrioriDetermination of Sample Grab

Size

To insure that the measurement precision is not due tosampling (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 samplingRSD,

Or, in terms of the measurement standard deviation