In probability theory and statistics, standard deviation is a measure of the variability or dispersion of a population, a data set, or a probability distribution. A low standard deviation indicates that the data points tend to be very close to the same value (the mean), while high standard deviation indicates that the data are “spread out” over a large range of values. For example, the average height for adult men in the United States is about 70 inches (180 cm), with a standard deviation of around 3 inches.

This means that most men (about 68%, assuming a normal distribution) have a height within 3 inches of the mean (67 inches (170 cm) – 73 inches (190 cm) inches), while almost all men (about 95%) have a height within 6 inches of the mean (64 inches (160 cm) – 76 inches (190 cm)). If the standard deviation were zero, then all men would be exactly 70 inches (180 cm) high. If the standard deviation were 20 inches (51 cm), then men would have much more variable heights, with a typical range of about 50 inches (130 cm) to 90 inches (230 cm).

In addition to expressing the variability of a population, standard deviation is commonly used to measure confidence in statistical conclusions. For example, the margin of error in polling data is determined by calculating the expected standard deviation in the results if the same poll were to be conducted multiple times. (Typically the reported margin of error is about twice the standard deviation, the radius of a 95% confidence interval.) In science, researchers commonly report the standard deviation of experimental data, and only effects that fall far outside the range of standard deviation are considered statistically significant—normal random error or variation in the measurements is in this way distinguished from causal variation.

Standard deviation is also important in finance, where the standard deviation on the rate of return on an investment is a measure of the risk. The term standard deviation was first used in writing by Karl Pearson in 1894 following use by him in lectures. This was as a replacement for earlier alternative names for the same idea: for example Gauss used "mean error".

A useful property of standard deviation is that, unlike variance, it is expressed in the same units as the data.

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