Standard deviation — Mathematics, 14–17 years
The mean gives a centre, but not how tightly data gather around it. Standard deviation measures the typical distance from the mean, helping you compare the spread of different data sets.
A measure of spread
Standard deviation describes how far data values typically sit from their mean. A small value means the data cluster closely; a large value means they are more spread out. It uses the original unit, such as seconds, euros or centimetres.
Why the mean is not enough
Two groups can have the same mean but very different consistency. A coach comparing race times needs to know whether athletes finish close together or vary widely, not just their average. Standard deviation was developed to turn that spread into one comparable number.
A small calculation
For 2, 4 and 6, the mean is 4. The differences are −2, 0 and 2; square them to get 4, 0 and 4. Their mean is 8/3, so the population standard deviation is √(8/3), about 1.63. The values usually lie about 1.63 units from the mean.
Ignoring the signs correctly
Adding the raw differences from the mean always gives zero, because values above and below the mean cancel. That makes it look as if there is no spread, which is a reasonable conclusion from an ordinary sum. Squaring first removes the signs while keeping every distance involved.
Comparing reliability
Standard deviation is used when averages alone hide variation: checking sensor readings, comparing delivery times or studying test scores. A low spread can mean a process is consistent, but it is not automatically better; the target and the size of the measurements still matter.
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