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Chapter 4 Evaluating Analytical Data 69

pected uncertainty for an analysis. Comparing the expected uncertainty to that
which is actually obtained can provide useful information. For example, in de-
termining the mass of a penny, we estimated the uncertainty in measuring mass
as ±0.002 g based on the balance’s tolerance. If we measure a single penny’s mass
several times and obtain a standard deviation of ±0.020 g, we would have reason
to believe that our measurement process is out of control. We would then try to
identify and correct the problem.
A propagation of uncertainty also helps in deciding how to improve the un-
certainty in an analysis. In Example 4.7, for instance, we calculated the concen-
tration of an analyte, obtaining a value of 126 ppm with an absolute uncertainty
of ±2 ppm and a relative uncertainty of 1.6%. How might we improve the analy-
sis so that the absolute uncertainty is only ±1 ppm (a relative uncertainty of
0.8%)? Looking back on the calculation, we find that the relative uncertainty is
determined by the relative uncertainty in the measured signal (corrected for the
reagent blank)
.
0 028
=± 0 0012, ± 0 12%
or
.
.
.
23 41
and the relative uncertainty in the method’s sensitivity, k,
.
0 003
=± 0 016, ± 1 6%
.
.
or
0 186
.
Of these two terms, the sensitivity’s uncertainty dominates the total uncertainty.
Measuring the signal more carefully will not improve the overall uncertainty
of the analysis. On the other hand, the desired improvement in uncertainty
can be achieved if the sensitivity’s absolute uncertainty can be decreased to
–1
±0.0015 ppm .
As a final example, a propagation of uncertainty can be used to decide which
of several procedures provides the smallest overall uncertainty. Preparing a solu-
tion by diluting a stock solution can be done using several different combina-
tions of volumetric glassware. For instance, we can dilute a solution by a factor
of 10 using a 10-mL pipet and a 100-mL volumetric flask, or by using a 25-mL
pipet and a 250-mL volumetric flask. The same dilution also can be accom-
plished in two steps using a 50-mL pipet and a 100-mL volumetric flask for the
first dilution, and a 10-mL pipet and a 50-mL volumetric flask for the second di-
lution. The overall uncertainty, of course, depends on the uncertainty of the
glassware used in the dilutions. As shown in the following example, we can use
the tolerance values for volumetric glassware to determine the optimum dilution
strategy. 5

4 9
EXAMPLE .
Which of the following methods for preparing a 0.0010 M solution from a
1.0 M stock solution provides the smallest overall uncertainty?

(a) A one-step dilution using a 1-mL pipet and a 1000-mL volumetric
flask.
(b) A two-step dilution using a 20-mL pipet and a 1000-mL volumetric flask
for the first dilution and a 25-mL pipet and a 500-mL volumetric flask for
the second dilution.