Standard deviation from values and frequencies

Standard deviation measures spread about the mean. Squaring deviations prevents positive and negative differences cancelling; averaging then taking a square root returns to the original units. Use the population standard deviation for the given data set.

\(\sigma=\sqrt{\dfrac{\sum x^2}{n}-\overline{x}^{\,2}}\), or with frequencies, \(\sigma=\sqrt{\dfrac{\sum fx^2}{\sum f}-\overline{x}^{\,2}}\). Here \(fx^2\) means frequency multiplied by the squared value, not \((fx)^2\).

Example

For \(2,4,4,6\), \(n=4\), \(\sum x=16\), \(\sum x^2=72\), so \(\overline{x}=4\) and \(\sigma=\sqrt{\dfrac{72}{4}-16}=\sqrt2\approx1.41\).

The frequency table \(x:2,4,6\), \(f:1,2,1\) represents the same data: \(\sum f=4\), \(\sum fx=16\), \(\sum fx^2=72\), giving the same results.

For interval-grouped data use class midpoints for \(x\). In the time example with midpoints \(5,15,25\) and frequencies \(4,8,8\), \(\sum fx^2=6900\), mean \(=17\), and estimated \(\sigma=\sqrt{\dfrac{6900}{20}-17^2}=\sqrt{56}\approx7.48\text{ min}\).

On a calculator enter frequencies as well as values, check the total count, and select population \(\sigma\), not the sample statistic dividing by \(n-1\). Keep full precision until the final result. Midpoint-based mean and standard deviation are estimates.