| Consider a sponge cut into a block of length, width and height. If
you were to make several measurements of the length with a precise electronic
vernier caliper, you would have a distribution of values that represented
the roughness and unevenness of the surfaces at each end. The results would
also depend upon the pressure that you applied to the sponge and may vary
with other variables such as mass of absorbed water. The length itself
is not perfectly defined and has an uncertainty which you determine by
repeated measurements at different positions and pressures etc.
The mean of a set of two or more measurements is their sum divided by
the number of measurements:
Lave = Sum(Li)/N
We define the standard deviation or uncertainty of a set of two or more
measurements to be the square root of the average square deviation from
the mean
sL
= (Sum(L-Lave)2/N)1/2
or root mean square deviation from the
mean. In student labs we often take two independent measurements and call
the uncertainty half the difference. If the number of measurements, N,
were much larger than two, the standard deviation of the mean would be
smaller than than that of individual measurements by the square root of
N-1. Thus the mean of one hundred measurements is about ten times more
accurate than any one of them and a single measurement has infinite uncertainy.
With this many measurements you might keep two significant figures in the
uncertainty rather than one.
If the block above were made of precision
machined steel and polished square within a wavelength of light and measured
with a meter stick of smallest increment equal to one millimeter, then
you might expect repeated measments to all be the same. But the uncertainty
is not zero as predicted by the the equation above since the instrument
is limited in its accuracy by the smallest division, or least count. The
human eye can easily resolve a tenth of a millimeter and we usually estimate
the next decimal place. The total uncertainty is the sum of the uncertainties
due to the object, the instrument and the observer. With all these
sources of uncertainty, a good measurement requires many observations which
vary over the object, the instrument and different observers.
What if we wanted to know the volume
of the sponge above after carefully measuring each dimension? Since uncertainties
are usually small, we can differentiate the equation for volume
V = LWH
to tell us what its uncertainty is due
to the three measurements. The total differential change in V is the sum
of contributions from all measured variables:
dV = Sum( (dV/dx) dx)
and for the sponge above
dV = WH dL + LH dW + LW dH
by the product rule. You can envision
these little volumes as extensions to the original volume in each of the
three axes. In this case we can divide the differential volume by the volume
itself and find that
dV/V = dL/L + dW/W + dH/H.
The total percentage uncertainty in
the volume is the sum of the percentage uncertainties in each measurement.
This equation should be modified due to the fact that uncertainties are
random and not in the same direction at the same time. The correct expression
for total percentage uncertainty is the hypotenuse of the three percents,
or square root of the sum of the squares. Since we usually keep only one
significant figure in uncertainty, corresponding to the last significant
figure of the mean, we usually just keep only the largest percentage uncertainty
and drop the others as negligible. You should check your understanding
of the propogation of uncertainties by showing that the percentage uncertainty
in the volume of a sphere is three times the percentage uncertainty of
its measured diameter. |
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