analysis and sample variances
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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
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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
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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 -
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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