Errors and Statistics

Instrument Uncertainty and Least Count


Each instrument has an inherent amount of uncertainty in its measurement. Even the most precise measuring device cannot give the actual value because to do so would require an infinitely precise instrument. A measure of the accuracy of an instrument is given by its uncertainty. As a good rule of thumb, the uncertainty of a measuring device is 50% of the least count. Recall that the least count is the smallest subdivision given on the measuring device. The uncertainty of the measurement should be given with the actual measurement, for example, 41.64 ± 0.05 cm.

Here are some typical uncertainties of various laboratory instruments:

Here's an example. The uncertainty of all measurements made with a metre stick whose smallest division (or least count) is one millimetre is 50% of 1mm or 0.05cm. Say you use that metre stick to measure a metal rod and find that the rod is between 10.2 cm and 10.3cm. You may think that the rod is closer to 10.2cm than it is to 10.3cm, so you make your best guess that the rod is, say, 10.23cm in length. Since the uncertainty in the measurement is 0.05cm, you would report the length of the metal rod to be 10.23 ± 0.05cm (0.1023 ± 0.0005 m).

With any experiment it is important to properly display the precision with which each measurement is made. No measurement is absolutely precise. For example, it is impossible to measure the exact length of an object. We might measure the length as 1.23cm, but this does not mean that the actual measurement is 1.23000000...cm! We must carefully describe how precise our measurement is. A experimental value of 1.23 ± 0.10 cm is less precise than a measurement of 1.23 ± 0.01cm. The ± term gives the measure of the precision of the measurement. The accuracy of the value is given either by percentage error or percentage difference.

When a quantity is graphed, it is common for the uncertainty of that quantity to be represented by error bars. For more information about error bars, see the Excel tutorial on using error bars.

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