Page 204 - Modern Analytical Chemistry

1400-CH07 9/8/99 4:03 PM Page 187

Chapter 7 Obtaining and Preparing Samples for Analysis 187
7B.3 How Much Sample to Collect

To minimize sampling errors, a randomly collected grab sample must be of an ap-
propriate size. If the sample is too small its composition may differ substantially
from that of the target population, resulting in a significant sampling error. Samples
that are too large, however, may require more time and money to collect and ana-
lyze, without providing a significant improvement in sampling error.
As a starting point, let’s assume that our target population consists of two types
of particles. Particles of type A contain analyte at a fixed concentration, and type B
particles contain no analyte. If the two types of particles are randomly distributed,
then a sample drawn from the population will follow the binomial distribution.* If
we collect a sample containing n particles, the expected number of particles con-
taining analyte, n A , is
n A = np
where p is the probability of selecting a particle of type A. The sampling standard
deviation is
s s = np(1 - p) 7.3

The relative standard deviation for sampling, s s,r , is obtained by dividing equation
7.3 by n A .
np(1 - p)
s s,r =
np
Solving for n allows us to calculate the number of particles that must be sampled to
obtain a desired sampling variance.

1 - p 1
n = ´ 7.4
2
p s s,r
Note that the relative sampling variance is inversely proportional to the number of
particles sampled. Increasing the number of particles in a sample, therefore, im-
proves the sampling variance.

7
EXAMPLE .4
Suppose you are to analyze a solid where the particles containing analyte
–7
represent only 1 ´10 % of the population. How many particles must be
collected to give a relative sampling variance of 1%?
SOLUTION
–7
Since the particles of interest account for 1 ´10 % of all particles in the
–9
population, the probability of selecting one of these particles is only 1 ´10 .
Substituting into equation 7.4 gives
-9
1 - 1 ´10 ) 1
(
n = -9 ´ 2 =1 ´10 13
001)
1 ´10 (.
Thus, to obtain the desired sampling variance we need to collect 1 ´10 13
particles.

*See Chapter 4 to review the properties of a binomial distribution.