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10

Precision of Calculated Values

KEY WORDS additive error, alkalinity, calcium, corrosion, dilution, error transmission, glassware,
hardness, Langlier stability index, LSI, measurement error, multiplicative error, precision, propagation
of variance, propagation of error, random error, relative standard deviation, RSI, Ryzner stability index,
saturation index, standard deviation, systematic error, titration, variance.

Engineers use equations to calculate the behavior of natural and constructed systems. An equation’s
solid appearance misleads. Some of the variables put into the equation are measurements or estimates,
perhaps estimated from an experiment or from experience reported in a handbook. Some of the constants
in equations, like π, are known, but most are estimated values. Most of the time we ignore the fact that
the calculated output of an equation is imprecise to some extent because the inputs are not known with
certainty.
In doing this we are speculating that uncertainty or variability in the inputs will not translate into
unacceptable uncertainty in the output. There is no need to speculate. If the precision of each measured
or estimated quantity is known, then simple mathematical rules can be used to estimate the precision of
the final result. This is called propagation of errors. This chapter presents a few simple cases without
derivation or proof. They can be derived by the general method given in Chapter 49.

Linear Combinations of Variables
The variance of a sum or difference of independent quantities is equal to the sum of the variances. The
measured quantities, which are subject to random measurement errors, are a, b, c,…:

y = a ++ + …
c
b
σ y = σ a + σ b + σ c + …
2
2
2
2
2
σ y = σ a + σ b + σ c + …
2
2
The signs do not matter. Thus, y = a − b − c − … also has σ y = σ a + σ b + σ c + …
2
2
2
2
We used this result in Chapter 2. The estimate of the mean is the average:
--- y 1 +(
y = 1 y 2 + y 3 + … + y n )
n
The variance of the mean is the sum of the variances of the individual values used to calculate the average:
1
σ y = ----- σ 1 +( 2 σ 2 + σ 3 + … + σ n )
2
2
2
2
n 2
Assuming that σ 1 = σ 2 = … = σ n = σ y , σ y = σ y , this is the standard error of the mean.
-------
n
© 2002 By CRC Press LLC