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 percent error or percent difference.

To find the uncertainty in our measurements, we will often calculate the standard deviation, or , of the measured value. Standard deviation is a measure of the variation of N data points (x₁...x_N) about an average value, , and is typically called the uncertainty in a measured result.

To calculate the average or mean value, , of a set of N measurements is

(1)

Once the mean value of the measurements is determined, it is helpful to define how much the individual measurements are scattered around about the mean. The deviation, , of any measurement, , from the mean is given by

(2)

Since the deviation may be either positive or negative, it is often more useful to use the mean deviation, or , to determine the uncertainty of the measurement. This is found by averaging the absolute deviations, ; that is,

(3)

It is common to report the experimental value, , of a measurement as

(4)

where , gives the measure of the precision of the measurement.

To avoid the use of absolute values we can use the square of the deviation, , to more accurately determine the uncertainty of our measurement. The standard deviation, , (sometimes called the root-mean square) is given by

(5)

(It can be shown that for a small number of measurements, Equation 5 becomes

(6)

where N is replaced by N - 1. Your instructor may want you to use this formula instead of Equation 5.)

Finally, the experimental result, , can then be written as

(7)

where , gives the measure of the precision of the measurement. Often scientists use the value of the standard deviation to serve as their data's Error bars.

Notice the standard deviation is always positive and has the same units as the mean value. It can be shown that there is a 68% likelihood that an individual measurement will fall within one standard deviation () of the true value. Furthermore, it can be shown that there also exists a 95% likelihood that an individual measurement will fall within two standard deviations () of the true value, and a 99.7% likelihood that it will fall within () of the true value.