PHYS101: Introduction to Mechanics

Uncertainty exists in any measured quantity because measurements are always performed by a person or instrument. For example, if you are using a ruler to measure length, it is necessary to interpolate between gradations given on the ruler. This gives the uncertain digit in the measured length. While there may not be much deviation, what you estimate to be the last digit may not be the same as someone else's estimation. We need to account for this uncertainty when we report measured values.

When measurements are repeated, we can gauge their accuracy and precision. Accuracy tells us how close a measurement is to a known value. Precision tells us how close repeat measurements are to each other. Imagine accuracy as hitting the bullseye on a dartboard every time, while precision corresponds to hitting the "triple 20" consistently. Another example is to consider an analytical balance with a calibration error so that it reads 0.24 grams too high. Although measuring identical mass readings of a single sample would mean excellent precision, the accuracy of the measurement would be poor.

To account for the uncertainty inherent in any measured quantity, we report measured quantities using significant figures or sig figs, which are the number of digits in a measurement you report based on how certain you are of your measurement. Reporting sig figs properly is important, and we need to account for sig figs when performing mathematical calculations using measured quantities. There are rules for determining the number of sig figs in a given measured quantity. There are also rules for carrying sig figs through mathematical calculations.