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Johns Hopkins University
What is Engineering?
Michael Karweit
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v   [2 RH c r  R 2 hc  H c r 2  2 Rrh c  hc r 2 ] 
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+ HWl  HLw  LWh  Hlw  Lhw  Whl  lwh .
First, let's look at v statistically, i.e., what would be the average value of the error v if
measurements were taken many times and averaged. Using overbar notation, we obtain
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v   [2 RH c r  R 2 hc  H c r 2  2 R hc  hc r 2 ] 
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 HWl  HLw  LWh  H lw  Lhw  W hl  lwh .
If hc, r, l, w, and h are all independent random variables with zero mean, then we get
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v   H c r 2 . All the other terms average to zero.
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Thus, we have deduced an average expected error in v. But, suppose we can't take
a lot of sample measurements, and we would like to know what is the "worst case" error.
That's fairly simple. Suppose we can estimate the maximum possible error on each
measurement. Let these be labeled rmax, lmax, hc max, hmax, and wmax. Then
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v max   [2 RH c rmax  R 2 hc max  H c rmax
 2 Rhc max  hc max rmax
]
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 HWl max  HLwmax  LWhmax  Hl max wmax  Lhmax wmax  Whmax l max  l max hmax wmax
A simplification of this is to assume that rmax, lmax, hc max, hmax, and wmax are all very small
compared to R, L, Hc, H, and W. Then terms containing two or more of these maximum
errors will be much smaller than terms containing only one. Consequently, if we ignore
these smaller terms, vmax can be approximated as
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v max   (2 RH c rmax  R 2 hc max )  LWhmax  LHwmax  WHl max .
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There's a reason for making this simplification, even though it's only an approximation to
the maximum error in v. The reason is that this simplified expression is easily deduced
for any combination of measurements. This is the result that one would obtain by taking
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the total differential of the function V ( R, H c , L,W , H )   R 2 H c  LWH . The total
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total differential is defined like this: If F is a differentiable function depending on n
variables x1 , x 2 , , x n , then infinitesimal variations in F are determined by infinitesimal
variations in the xis as:
F
F
F
dF ( x1 , x 2 ,..., x n ) 
dx1 
dx 2   
dx
 x1
 x2
 xn n
dF is called the “total differential” . Our variations rmax, lmax, hc max, hmax, and wmax are not
infinitesimal. But, if they're quite small, then the total differential is a pretty good
approximation to the total error vmax (that is, dF in the notation immediately above).
4/23/01
Uncertainty
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