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Chapter 4 Evaluating Analytical Data 65

It is easy to see that combining uncertainties in this way overestimates the total un-
certainty. Adding the uncertainty for the first delivery to that of the second delivery
assumes that both volumes are either greater than 9.992 mL or less than 9.992 mL.
At the other extreme, we might assume that the two deliveries will always be on op-
posite sides of the pipet’s mean volume. In this case we subtract the uncertainties
for the two deliveries,
(9.992 mL + 9.992 mL) ± (0.006 mL – 0.006 mL) = 19.984 ± 0.000 mL

underestimating the total uncertainty.
So what is the total uncertainty when using this pipet to deliver two successive
volumes of solution? From the previous discussion we know that the total uncer-
tainty is greater than ±0.000 mL and less than ±0.012 mL. To estimate the cumula-
tive effect of multiple uncertainties, we use a mathematical technique known as the
propagation of uncertainty. Our treatment of the propagation of uncertainty is
based on a few simple rules that we will not derive. A more thorough treatment can
be found elsewhere. 4

4 C.1 A Few Symbols

Propagation of uncertainty allows us to estimate the uncertainty in a calculated re-
sult from the uncertainties of the measurements used to calculate the result. In the
equations presented in this section the result is represented by the symbol R and the
measurements by the symbols A, B, and C. The corresponding uncertainties are s R ,
s A , s B , and s C . The uncertainties for A, B, and C can be reported in several ways, in-
cluding calculated standard deviations or estimated ranges, as long as the same form
is used for all measurements.

4 C.2 Uncertainty When Adding or Subtracting
When measurements are added or subtracted, the absolute uncertainty in the result
is the square root of the sum of the squares of the absolute uncertainties for the in-
dividual measurements. Thus, for the equations R = A + B + C or R = A + B – C, or
any other combination of adding and subtracting A, B, and C, the absolute uncer-
tainty in R is

s R = s 2 s + 2 s + 2 4.6
A B C

4 5
EXAMPLE .
The class A 10-mL pipet characterized in Table 4.8 is used to deliver two
successive volumes. Calculate the absolute and relative uncertainties for the
total delivered volume.
SOLUTION
The total delivered volume is obtained by adding the volumes of each delivery;
thus
V tot = 9.992 mL + 9.992 mL = 19.984 mL
Using the standard deviation as an estimate of uncertainty, the uncertainty in
the total delivered volume is
0
s R = (.006 ) 2 +( .006 ) 2 = .0085
0
0