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L1592_Frame_C10 Page 90 Tuesday, December 18, 2001 1:46 PM
Likewise, if y = a/b, the variance is:
2 a
σ y = σ a a + σ b ----- 2
2
2
2
b 2
and
2 2 2
------ = ------ + σ b
σ a
σ y
------
y 2 a 2 b 2
Notice that each term is the square of the relative standard deviation (RSD) of the variables. The RSDs
are σ y /y, σ a /a, and σ b /b.
These results can be generalized to any combination of multiplication and division. For:
y = kab/cd
where a, b, c and d are measured and k is a constant, there is again a relationship between the squares
of the relative standard deviations:
----- = σ a 2 + σ b 2 + 2 + σ d 2
σ y
σ c
-----
-----
-----
-----
y a b d
c
Example 10.4
X a V
The sludge age of an activated sludge process is calculated from θ = ------------- , where X a is mixed-
Q w X w
liquor suspended solids (mg/L), V is aeration basin volume, Q w is waste sludge flow (mgd), and
X w is waste activated sludge suspended solids concentration (mg/L). Assume V = 10 million
gallons is known, and the relative standard deviations for the other variables are 4% for X a , 5%
for X w , and 2% for Q w . The relative standard deviation of sludge age is:
σ θ
----- = 4 + 5 + 2 = 45 = 6.7%
2
2
2
θ
The RSD of the final result is not so much different than the largest RSD used to calculate it.
This is mainly a consequence of squaring the RSDs.
Any efforts to improve the precision of the experiment need to be directed toward improving the
precision of the least precise values. There is no point wasting time trying to increase the precision of
the most precise values. That is not to say that small errors are unimportant. Small errors at many stages
of an experiment can produce appreciable error in the final result.
Error Suppression and Magnification
A nonlinear function can either suppress or magnify error in measured quantities. This is especially true
of the quadratic, cubic, and exponential functions that are used to calculate areas, volumes, and reaction
rates in environmental engineering work. Figure 10.1 shows that the variance in the final result depends
on the variance and the level of the inputs, according to the slope of the curve in the range of interest.
© 2002 By CRC Press LLC
