Significant Figures:
As a scientist or engineer we deal with analytically derived values
obtained on digital equipment that reports the maximum number of
digits that will fit on the display, typically 9. This is also
true when calculators are used to determine a value, I will drive to
Columbus at 60 miles per hour and it is about 130 miles, how long
will it take me? 2.166666667 hours.
When reporting a value we need to consider the reasonableness of
what we report. The last digit in the time reported above is
10-7 seconds which is a time that could possibly be measured but has
no relevance to the problem and is certainly beyond the one digit of
accuracy given by the speed. If you reported this result you
could expect to be ridiculed. The correct value to report is
two hours. This is determined by the least accurate value used
in the calculation and because the term "about 130 miles" implies
some uncertainty as to the distance on the order of 10% or
higher. Reporting the correct number of significant figures
involves consideration of the error in the value and the
reasonableness of the last significant figure reported in the
context of the situation.
We measure the glass transition for polystyrene at 121.54°C in a
differential scanning calorimeter by observing a sigmoidal shaped
feature in the scan and performing a computer based determination of
the midpoint of the decay in heat flow. The analysis has a
number of potential pitfalls. The value will depend on how the
baseline for the measurement is determined for instance. This
can shift the value by 5 to 10 °C. Further, the heating
rate changes the Tg value by ±10 °C. The thermal
history of the sample can have similar effects on the observed
value. In this case the analytic fit to the data can generate
an error of ±0.05°C but the actual accuracy is on the
order of ±10°C or more. So reporting more than two
significant figures is misleading.
The main issue with significant figures and error analysis is that
you need to develop a level of comfort with the values that you
report so that you can standby your results. When you report a
value you report both the number and your confidence in that number
as reflected by the significant figures that you report and/or an
explicit statement of the error that you believe is associated with
the value.