I had stayed out of the last round of this discussion, but since I
have now entered it, let me give one more reply. I will restate the
situation presented by Ken Fox with symbols that work in ASCII-text-only email.
L = 3.75 cm, dL =.05 cm
W = 1.32 cm, dW = .05 cm
H = 0.54 cm, dH = .05 cm
Therefore the volume, as Ken noted, is: V = 2.67300 cm^3.
The question is how should this number be reported and with what uncertainty.
As D. C. Baird points out in his book, _Experimentation: An
Introduction to Measurement Theory and Experimental Design_
(Prentice-Hall), the formal uncertainty for a function such as V = L
W H is dV = W H dL + H L dW + L W dH.
This gives: dV = 0.384 cm^3 to 3 digits.
As Ken notes the rules of significant digits would lead us to round this to:
V = 2.7 cm^3, dV = 0.4 cm^3. And he was not happy with this.
However, let us take a very liberal view and say that it is known
that the uncertainties in L, W, and H are not 0.06 cm and not 0.07
cm, but they are in the range between 0.045 and 0.055 cm.
Doing the error calculation with these outer bounds gives:
dV = 0.346 cm^3, for dL=dW=dH=0.045 cm
dV = 0.423 cm^3, for dL=dW=dH=0.055 cm
Thus, from my figures, we cannot tell the dV to more than one digit
which is what is given by the rules of significant digits. And with
an error of +/- 0.4 cm^3, I don't see how one can report the volume
to better than two digits (V=2.7 cm^3) as would be suggested by the
rules of significant digits. In fact the fuller error analysis tells
us that even the rules of significant digits may have given us a bit
more precision than we really have. But here at least, these rules of
significant digits appear to serve us much better than our gut
feelings. And, in my opinion, that is their purpose.
Richard Bowman
Dept. of Physics
Bridgewater College, Bridgewater, VA, USA
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