Definition: The standard deviation measures how far data values typically fall from the mean; a small value means data cluster tightly and a large value means they spread out.
What does it measure?
Standard deviation is a kind of typical distance from the mean, in the same units as the data. To compute it: find the mean, subtract it from each value, square the deviations, add, divide, and take a square root. Divide by n – 1 for a sample and by n for a whole population.
How does it look in a small data set?
Worked example: the data are 4, 8, 10, 14. The mean is 9, and the deviations are -5, -1, 1, 5. Squared they give 25, 1, 1, 25, total 52.
- Sample: s = square root of (52/3), about 4.16
- Population: sigma = square root of (52/4) = square root of 13, about 3.61
What do students get wrong?
Skipping the squaring step, which makes the deviations add to zero. Another trap is thinking the standard deviation can be negative; it is zero or positive. Mathbot checks that it is smaller than the range.
Where to learn it
- Exploring One-Variable Data: Shape, Center, Spread – lesson
- Normal Curves, z-Scores and Study Design – lesson