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Unformatted text preview: very careful to estimate or eliminate (by
other means) systematic uncertainties well because 243 APPENDIX: ACCURACY, PRECISION AND UNCERTAINTY measurement could be off is a fraction of a mm. To
be more precise, we can estimate it to be about a third
of a mm, so we can say that the length of the key is
5.37 ± 0.03 cm. they cannot be eliminated in this way; they would
just shift the distributions in Figure 1 left or right.
Roughly speaking, the average or “center” of the
distribution is the “measurement,” and the width or
“deviation” of the distribution is the random
uncertainty. Another time you may need to estimate uncertainty
is when you analyze video data. Figures 3 and 4
show a ball rolling off the edge of a table. These are
two consecutive frames, separated in time by 1/30 of
a second. How do I determine the uncertainty?
This Appendix will discuss three basic techniques for
determining the uncertainty: estimating the
uncertainty, measuring the average deviation, and
finding the uncertainty in a linear fit. Which one
you choose will depend on your situation, your
available means of measurement, and your need for
precision. If you need a precise determination of
some value, and you are measuring it directly (e.g.,
with a ruler or thermometer), the best technique is to
measure that value several times and use the average
deviation as the uncertainty. Examples of finding the
average deviation are given below. Figure 3 How do I estimate uncertainties?
If time or experimental constraints make repeated
measurements impossible, then you will need to
estimate the uncertainty.
When you estimate
uncertainties you are trying to account for anything
that might cause the measured value to be different if
you were to take the measurement again. For
example, suppose you were trying to measure the
length of a key, as in Figure 2. Figure 4 Figure 2 The exact moment the ball left the table lies
somewhere between these frames. We can estimate
that this moment occurs midway between them (
1
t 10 60 s ). Since it must occur at some point If the true value were not as important as the
magnitude of the value, you could say that the key’s
length was 5cm, give or take 1cm. This is a crude
estimate, but it may be acceptable. A better estimate
of the key’s length, as you saw in Appendix A, would
be 5.37cm.
This tells us that the worst our between them, the worst our estimate could be off by 244 APPENDIX: ACCURACY, PRECISION AND UNCERTAINTY is 1
60 s . We can therefore say the time the ball leaves 1
the table is t 10 60 1
60 3.131 average
The deviations are: 0.009g, 0.002g, 0.013g,
0.013g, 0.005g, 0.006g
Sum of deviations: 0.048g
Average deviation:
(0.048g)/6= 0.008g
Mass of penny B: 3.131 ± 0.008g s. How do I find the average deviation?
If estimating the uncertainty is not good enough for
your situation, you can experimentally determine the
uncertainty by making several measurements and
calculating the average deviation of those
measurements. To find the average deviation: (1)
Find the average of all your measurements; (2) Find
the absolute value of the difference of each
measurement from the average (its deviation); (3)
Find the average of all the deviations by adding them
up and dividing by the number of measurements. Of
course you need to take enough measurements to get
a distribution for which the average has some
meaning. Finding the Uncertainty in a Linear Fit
Sometimes, you will need to find the uncertainty in a
linear fit to a large number of measurements. The
most common situation like this that you will
encounter is fitting position or velocity with respect
to time from MotionLab.
When you fit a line to a graph, you will be looking
for the “best fit” line that “goes through the middle”
of the data; see the appendix about graphs for more
about this procedure. To find the uncertainty, draw
the lines with the greatest and least slopes that still
roughly go through the data. These will be the upper
and lower limits of the uncertainty in the slope.
These lines should also have lesser and greater yintercepts than the “best fit” line, and they define the
lower and upper limits of the uncertainty in the yintercept. In example 1, a class of six students was asked to find
the mass of the same penny using the same balance.
In example 2, another class measured a different
penny using six different balances. Their results are
listed below:
Class 1: Penny A massed by six different students on the
same balance.
Mass (grams)
3.110
3.125
3.120
3.126
3.122
3.120
3.121 average.
The deviations are: 0.011g, 0.004g, 0.001g,
0.005g, 0.001g, 0.001g
Sum of deviations: 0.023g
Average deviation:
(0.023g)/6 = 0.004g
Mass of penny A: 3.121 ± 0.004g Note that when you do this, the uncertainties above
and below your “best fit” values will, in general, not
be the same; this is different than the other two
methods we have presented.
For example, in Figure 5, the yintercept is 4.25
+2.75/2.00, and the slope is 0.90 +0.20/0.25.
Figure 5a Class 2: Penny B massed by...
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 Spring '14

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