Page 209 - Modern Analytical Chemistry

1400-CH07 9/8/99 4:03 PM Page 192

192 Modern Analytical Chemistry

This is not an uncommon problem. For a target population with a relative sampling
variance of 50 and a desired relative sampling error of ±5%, equation 7.7 predicts
that ten samples are sufficient. In a simulation in which 1000 samples of size 10
were collected, however, only 57% of the samples resulted in sampling errors of less
8
than ±5%. By increasing the number of samples to 17 it was possible to ensure that
the desired sampling error was achieved 95% of the time.
7B.5 Minimizing the Overall Variance

A final consideration in developing a sampling plan is to minimize the overall vari-
ance for the analysis. Equation 7.2 shows that the overall variance is a function of
the variance due to the method and the variance due to sampling. As we have seen,
we can improve the variance due to sampling by collecting more samples of proper
size. Increasing the number of times we analyze each sample improves the variance
2
2
due to the method. If s s is significantly greater than s m , then the method’s variance
can be ignored and equation 7.7 used to estimate the number of samples to analyze.
Analyzing any sample more than once will not improve the overall variance, since
the variance due to the method is insignificant.
2
2
If s m is significantly greater than s s , then we only need to collect and analyze a
single sample. The number of replicate analyses, n r , needed to minimize the error
due to the method is given by an equation similar to equation 7.7
22
ts m
n r =
e 2
Unfortunately, the simple situations just described are often the exception. In
many cases, both the sampling variance and method variance are significant, and
both multiple samples and replicate analyses of each sample are required. The over-
all error in this circumstance is given by
/
12
s æ 2 s s m ö
2
e = t ç + ÷ 7.8
sr
è n s nn ø
Equation 7.8 does not have a unique solution because different combinations of n s
and n r give the same overall error. The choice of how many samples to collect and
how many times each sample should be analyzed is determined by other concerns,
such as the cost of collecting and analyzing samples, and the amount of available
sample.

7 9
EXAMPLE .
A certain analytical method has a relative sampling variance of 0.40% and a
relative method variance of 0.070%. Evaluate the relative error (a= 0.05) if
(a) you collect five samples, analyzing each twice; and, (b) you collect two
samples, analyzing each five times.
SOLUTION
Both sampling strategies require a total of ten determinations. Using Appendix 1B,
we find that the value of t is 2.26. Substituting into equation 7.8, we find that the
relative error for the first sampling strategy is
12
/
.
æ 040. 0 070 ö
e = 226 ç + ÷ = 067.%
.
52)
è 5 () ( ø