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The random errors in filling a 250-mL flask might be ±0.05 mL, or only 0.02% of the total volume
of the flask. The random error in filling a transfer pipette should not exceed 0.006 mL, giving an error
of about 0.024% of the total volume (Miller and Miller, 1984). The error in reading a burette (of the
conventional variety graduated in 0.1-mL divisions) is perhaps ±0.02 mL. Each titration involves two
such readings (the errors of which are not simply additive). If the titration volume is about 25 mL, the
percentage error is again very small. (The titration should be arranged so that the volume of titrant is
not too small.)
In skilled hands, with all precautions taken, volumetric analysis should have a relative standard
deviation of not more than about 0.1%. (Until recently, such precision was not available in instrumental
analysis.)
Systematic errors can be due to calibration, temperature effects, errors in the glassware, drainage
errors in using volumetric glassware, failure to allow a meniscus in a burette to stabilize, blowing out
a pipette that is designed to drain, improper glassware cleaning methods, and “indicator errors.” These
are not subject to prediction by the propagation of error formulas.

Comments
The general propagation of error model that applies exactly to all linear models z = f(x 1 , x 2 ,…, x n ) and
approximately to nonlinear models (provided the relative standard deviations of the measured variables
are less than about 15%) is:

σ z ≈ ( ∂z/∂x 1 ) σ 1 + ( ∂z/∂x 2 ) σ 2 + … + ( ∂z/∂x n ) σ n 2
2
2
2
2
2
2
where the partial derivatives are evaluated at the expected value (or average) of the x i . This assumes that
there is no correlation between the x’s. We shall look at this and some related ideas in Chapter 49.
References

Betz Laboratories (1980). Betz Handbook of Water Conditioning, 8th ed., Trevose, PA, Betz Laboratories.
Langlier, W. F. (1936). “The Analytical Control of Anticorrosion in Water Treatment,” J. Am. Water Works
Assoc., 28, 1500.
Miller, J. C. and J. N. Miller (1984). Statistics for Analytical Chemistry, Chichester, England, Ellis Horwood
Ltd.
Ryznar, J. A. (1944). “A New Index for Determining the Amount of Calcium Carbonate Scale Formed by
Water,” J. Am. Water Works Assoc., 36, 472.
Spencer, G. R. (1983). “Program for Cooling-Water Corrosion and Scaling,” Chem. Eng., Sept. 19, pp. 61–65.

Exercises

10.1 Titration. A titration analysis has routinely been done with a titrant strength such that con-
centration is calculated from C = 20(y 2 − y 1 ), where (y 2 − y 1 ) is the difference between the final
and initial burette readings. It is now proposed to change the titrant strength so that C =
40(y 2 − y 1 ). What effect will this have on the standard deviation of measured concentrations?
10.2 Flow Measurement. Two flows (Q 1 = 7.5 and Q 2 = 12.3) merge to form a larger flow. The
standard deviation of measurement on flows 1 and 2 are 0.2 and 0.3, respectively. What is
the standard deviation of the larger downstream flow? Does this standard deviation change
when the upstream flows change?

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