Question

The volume of a cylinder is V space equals space pi space r squared space h, where r is the radius of the circular faces and h is the height of the cylinder. Your measurements show that the mean value of r is 21 cm and its statistical and instrumental uncertainties turned out to be 0.10 cm and 0.20 cm, respectively. Likewise, that of h are 13 cm for the mean, 0.2 cm (statistical) and 0.3 cm (instrumental) for its uncertainties. What can you say about which source of uncertainty has the largest contribution to the overall uncertainty of V ? What would you need to do to improve (reduce) the uncertainty on your calculation of V?

Answer 1

Uncertainty is a quantification of the doubt about the measurement result. The uncertainty estimate associated with a measurement should account for both the accuracy and precision of the measurement.

Statistical (Random) Error: The statistical uncertainty of a measurement is the uncertainty that reflects the fact that every time you make a measurement, you measure a slightly different quantity each time. The tendency for a measured value to "jump around" from measurement to measurement is the statistical error. Here, statistical error in r and h are 0.10 cm and 0.2 cm.

Systematic Error: This is uncertainty and error in your measurement caused by anything that is not statistical uncertainty. This includes instrumental effects, note-taking things into account and gross errors. Here, instrumental errors in r and h are 0.20 cm and 0.3 cm.

Precision is often reported quantitatively by using relative or fractional uncertainty:

fractional overall uncertainties will depend more on the uncertainties in r as it is multiplied by a factor of 2 in the above equation. But, if you put the values of , you will get equal contribution from these two measurement quantities.

Always remember that it is usually as important to minimise uncertainties as it is to quantify them. There are some good practices which can help to reduce uncertainties in making measurements generally. A few recommendations are:

• Calibrate measuring instruments (or have them calibrated for you) and use the calibration corrections which are given on the certificate. In this way you can reduce the instrumental uncertainties. Here, you have to choose the best measuring instruments, and use calibration facilities with the smallest uncertainties to measure the length and radius.
• Use an uncertainty estimate as I did in the above equation to identify the worst uncertainties (here r), and address that carefully.