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Unformatted text preview: six different students on
six different balances
Mass (grams)
3.140
3.133
3.144
3.118
3.126
3.125 245 APPENDIX: ACCURACY, PRECISION AND UNCERTAINTY Figure 5b Figure 6 Mass of pennies (in grams) with uncertainties
An important application of this is determining
agreement between experimental and theoretical
values. If you use a formula to generate a theoretical
value of some quantity and use the method below to
generate the uncertainty in the calculation, and if you
generate an experimental value of the same quantity
by measuring it and use the method above to
generate the uncertainty in the measurement, you
can compare the two values in this way. If the ranges
overlap, then the theoretical and experimental values
agree. If the ranges do not overlap, then the
theoretical and experimental values do not agree. Figure 5c What are R2, X2, and p?
It sometimes happens in statistical analysis that
instead of determining whether two numbers agree,
you need to determine whether a function
(“theoretical value”) and some data (“experimental
value”) agree. Our method of comparing two
numbers with uncertainties is too primitive for this
task. R2 (the Pearson correlation), X2 (Greek letter
“Chi,” not Roman X), and p are numbers that
describe how well these things agree. They are too
sophisticated for this appendix, but you may see
them from time to time. If you feel comfortable with
some basic statistics, you can look them up. You
should never need to calculate them by hand; let
your fitting software do it for you if your analysis
gets that sophisticated. The most you might
encounter in this class is that spreadsheet programs
will give you R2 if you use them to fit data; for your
purposes, you can consider your fit “good” if
R2≥0.95. However you choose to determine the uncertainty,
you should always state your method clearly in your
report.
How do I know if two values are the same?
Go back to the pennies. If we compare only the
average masses of the two pennies we see that they
are different. But now include the uncertainty in the
masses. For penny A, the most likely mass is
somewhere between 3.117g and 3.125g. For penny B,
the most likely mass is somewhere between 3.123g
and 3.139g. If you compare the ranges of the masses
for the two pennies, as shown in Figure 6, they just
overlap. Given the uncertainty in the masses, we are
able to conclude that the masses of the two pennies
could be the same. If the range of the masses did not
overlap, then we ought to conclude that the masses
are probably different. Which result is more precise?
Suppose you use a meter stick to measure the length
of a table and the width of a hair, each with an
uncertainty of 1 mm. Clearly you know more about
the length of the table than the width of the hair.
Your measurement of the table is very precise but
your measurement of the width of the hair is rather 246 APPENDIX: ACCURACY, PRECISION AND UNCERTAINTY of penny B is different, since the range of the new
value does not overlap the range of the previous
value. However, that conclusion would be wrong
since our uncertainty has not taken into account the
inaccuracy of the balance. To determine the accuracy
of the measurement, we should check by measuring
something that is known. This procedure is called
calibration, and it is absolutely necessary for making
accurate measurements. crude. To express this sense of precision, you need to
calculate the percentage uncertainty. To do this,
divide the uncertainty in the measurement by the
value of the measurement itself, and then multiply by
100%. For example, we can calculate the precision in
the measurements made by class 1 and class 2 as
follows:
Precision of Class 1's value:
(0.004 g ÷ 3.121 g) x 100% = 0.1 %
Precision of Class 2's value:
(0.008 g ÷ 3.131 g) x 100% = 0.3 % Be cautious! It is possible to make measurements that are
extremely precise and, at the same time, grossly
inaccurate. Class 1's results are more precise. This should not be
surprising since class 2 introduced more uncertainty
in their results by using six different balances instead
of only one. How can I do calculations with values that have
uncertainty?
When you do calculations with values that have
uncertainties, you will need to estimate (by
calculation) the uncertainty in the result. There are
mathematical techniques for doing this, which
depend on the statistical properties of your
measurements. A very simple way to estimate
uncertainties is to find the largest possible uncertainty
the calculation could yield.
This will always
overestimate the uncertainty of your calculation,
but an overestimate is better than no estimate or an
underestimate.
The method for performing
arithmetic operations on quantities with uncertainties
is
illustrated
in
the
following
examples: Which result is more accurate?
Accuracy is a measure of how your measured value
compares with the real value. Imagine that class 2
made the measurement again using only one balance.
Unfortunately, they chose a balanc...
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 Spring '14

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