Representing Significant Digits
Introduction
You've collected ten measurements on the
distance a marble rolled in a Physics lab. You calculate the mean
distance and want to record this value in your lab. Do you record
it as:
112.3895 or
112.3 or
112?
This page will give you some guidance on how
to report your experimental results with the appropriate number
of significant digits.
When documenting experimental data, it is important
not to state the data with too much precision. That is, the number of
significant digits should reflect the level of experimental uncertainty.
How do you know what the level of experimental uncertainty and, therefore,
the precision of your measurements is? The best way is to take multiple
measurements and perform some basic statistical calculations. Short
of that, you can also make visual inspections of both the instrumentation
you are using and the data you have collected. First we will look at
visual inspection methods and then the statistical calculations.
Visual Inspection of Equipment
If you are recording data using analog equipment,
then you will typically be reading values from a dial, gauge, or other
scale that represents tick marks for the major or minor units. What
are the major or minor units on the scale? Is the space between the
smallest tick marks small enough that you can interpolate a fractional
distance between two marks (e.g., the pointer is one quarter the way
over to the next mark)? The smallest unit (or fraction of a unit) you
can read should give you an idea what the precision of your measurement
is. You should then use this as a guide for reporting any values derived
from subsequent calculations. For example, lets say that you’ve taken
six measurements that can be read to the nearest millimeter:
11, 12,
15, 14, 13, 14
It would not be appropriate to report the mean to three
significant digits: 13.167. Instead, you would probably want to use,
at most, the tenths digit: 13.2.
Visually inspecting the range of values
in your data set will also help you decide the number of significant
digits to display. For example, you’ve taken six measurements in centimeters:
12, 45, 89, 36, 22, 53
Given that the data is ranging not just in the
singles digit, but also in the tens digit (from 12 to 89), you would
want to report the mean as 43, not as 42.8.
When taking readings using
digital equipment, the readout will often give an unjustified confidence
in the precision of your readings. Say, for instance, you take repeated
measurements of voltages on a digital voltmeter:
2.3483, 3.2958, 1.8934,
5.3279, 4.4423
This is an unreasonable display of precision given your
values range between 1.9 and 5.3. While it is possible the voltmeter
can read to this level of precision, in theory, your experimental setup
is not allowing for this level of precision. A better representation
of this data would be:
2.3, 3.3, 1.9, 5.3, 4.4
Statistical Calculations
With some simple statistical calculations, you
can come up with a much better representation of the uncertainty in
your data. In addition to calculating the mean (i.e., average)
of a set of measurements, you can also calculate the standard deviation
of the measurements. The standard deviation (S.D.) characterizes
the average uncertainty of the measurements. Operationally, if you continued
to take measurements, approximately 68% of your measurements would fall
within a distance of one S.D. on either side of the mean of these measurements.
If we take the first data set above:
11, 12, 15, 14, 13, 14
If we continued
to take measurements, we could be pretty confident that 68% of the measurements
would fall within:
13.2 +/ 1.5 mm
Here, the uncertainty is represented
by the S.D.
When reporting the mean, it is often preferred
to give the standard deviation of the mean, or standard
error. The standard error (S.E.) reflects the number of measurements
that are taken and grows smaller as the number of measurements (N) grows
larger. For the same data set above, the uncertainty represented by
the S.E. would be:
13.2
+/ 0.6 mm
Now, getting back to significant digits. The general rule
is that experimental uncertainties should be rounded to one significant
digit. All of the following are appropriate notations:
13.2 +/ 0.6
mm
43 +/ 11 cm
8.76 +/ 0.02 volts
It follows that the reported value
of the mean (or other value) should be of the same order of magnitude
as the experimental uncertainty.
More Info
For more information on statistical calculations
and graphical display of uncertainty, see the graphing tutorial on Using
Error Bars in Graphs.
For more information on recording data in tables, see Designing Tables.
Reference
Taylor, J. R. (1997). Error analysis: The study of uncertainties in physical measurements (2nd ed.). Sausalito, CA: University Science Books.
