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.
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.
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
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 %
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.
Which result is more accurate?
Accuracy
is a measure of how your measured value
compares with the real value.
Imagine that class 2
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