10
Significant Figures and an Introduction to the Normal Distribution
Object
:
To become familiar with the proper use of significant figures and to become acquainted
with some rudiments of the theory of measurement.
Apparatus
:
Stopwatch, pendulum.
References
:
Mathematical Preparation for General Physics, Chapter 2, by Marion & Davidson;
Introduction to the Theory of Error by Yardley & Beers.
Significant Figures
Laboratory work involves the recording of various kinds of measurements and combining them to
obtain quantities that may be compared with a theoretical result.
For such a comparison to be meaningful,
the experimenter must have some idea of how accurate his measurements are, and should report this along
with his data and conclusion.
One obvious limit to attainable accuracy in any instance is the size of the smallest division of the
measuring instrument.
For example, with an ordinary meter stick, one can measure to within 1 mm.
Thus,
the result of such a measurement would be given as, e.g., 0.675 m.
This result contains three significant
figures.
It does not convey the same meaning as 0.6750 m which implies that a device capable of
measuring to 0.0001 m or 0.1 mm was used, and that the result was as given.
If the object were less than
0.1 m long, a measurement might give 0.075 m which contains only 2 significant figures because zeros used
to locate the decimal point are not significant figures.
For a still smaller object, a one significant figure
result might be obtained, e.g. 0.006 m.
It is important, in any kind of measurement, to judge all the factors
affecting the accuracy of that measurement and to record the data using the appropriate number of
significant figures.
When combining quantities, it is possible, using addition and subtraction, to get a result with more
or less significant figures than the original quantities had.
Thus,
0.721 m
+
0.675 m
1.396 m
this leads to a 4 significant figure result.
On the other hand
0.721 m
−
0.675 m
0.046 m
and one significant figure has been lost.
To add or subtract two quantities of different accuracy, the most
accurate must be rounded off.
For example, to add 12.3 and 1.57 m, one must first round off 1.57 m to 1.6
m and then add.
To multiply and divide, the rule is that the result must contain the same number of significant