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In the example, we used a standard deviation of 0.15 pH units for pH s . Let us apply the same error
propagation technique to see whether this was reasonable. To keep the calculations simple, assume that A,
K s , K 2 , and µ are known exactly (in reality, they are not). Then:

−2
2
−2
Var(pH s ) = (log 10 e) {[Ca] Var[Ca] + [Alk] Var[Alk]}
The variance of pH s depends on the level of the calcium and alkalinity as well as on their variances.
Assuming [Ca] = 36 mg/L, σ [Ca] = 3 mg/L, [Alk] = 50 mg/L, and σ [Alk] = 3 mg/L gives:

2
−2
−2
2
Var(pH s ) = 0.1886{[36] (3) + [50] (3) } = 0.002
which converts to a standard deviation of 0.045, much smaller than the value used in the earlier example.
Using this estimate of Var(pH s ) gives approximate 95% confidence intervals of:

0.03 < LSI < 0.47
6.23 < RSI < 6.77

This example shows how errors that seem large do not always propagate into large errors in calculated
values. But the reverse is also true. Our intuition is not very reliable for nonlinear functions, and it is
useless when several equations are used. Whether the error is magnified or suppressed in the calculation
depends on the function and on the level of the variables. That is, the final error is not solely a function
of the measurement error.

Random and Systematic Errors

The titration example oversimplifies the accumulation of random errors in titrations. It is worth a more
complete examination in order to clarify what is meant by multiple sources of variation and additive
errors. Making a volumetric titration, as one does to measure alkalinity, involves a number of steps:

1. Making up a standard solution of one of the reactants. This involves (a) weighing some solid
material, (b) transferring the solid material to a standard volumetric flask, (c) weighing the
bottle again to obtain by subtraction the weight of solid transferred, and (d) filling the flask
up to the mark with reagent-grade water.
2. Transferring an aliquot of the standard material to a titration flask with the aid of a pipette.
This involves (a) filling the pipette to the appropriate mark, and (b) draining it in a specified
manner into the flask.
3. Titrating the liquid in the flask with a solution of the other reactant, added from a burette. This
involves filling the burette and allowing the liquid in it to drain until the meniscus is at a constant
level, adding a few drops of indicator solution to the titration flask, reading the burette volume,
adding liquid to the titration flask from the burette a little at a time until the end point is adjudged
to have been reached, and measuring the final level of liquid in the burette.
The ASTM tolerances for grade A glassware are ±0.12 mL for a 250-mL flask, ±0.03 mL for a 25-mL
pipette, and ±0.05 mL for a 50-mL burette. If a piece of glassware is within the tolerance, but not exactly
the correct weight or volume, there will be a systematic error. Thus, if the flask has a volume of 248.9 mL,
this error will be reflected in the results of all the experiments done using this flask. Repetition will not
reveal the error. If different glassware is used in making measurements on different specimens, random
fluctuations in volume become a random error in the titration results.
© 2002 By CRC Press LLC