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.