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In probability and statistics, the standard deviation is the most common measure of statistical dispersion. Simply put, standard deviation measures how spread out the values in a data set are. More precisely, it is a measure of the average distance of the data values from their mean. If the data points are all close to the mean, then the standard deviation is low (closer to zero). If many data points are very different from the mean, then the standard deviation is high (further from zero). If all the data values are equal, then the standard deviation will be zero. The standard deviation has no maximum value although it is limited for most data sets.

The standard deviation is defined as the square root of the variance. This means it is the root mean square (RMS) deviation from the arithmetic mean. The standard deviation is always a positive number (or zero) and is always measured in the same units as the original data. For example, if the data are distance measurements in meters, the standard deviation will also be measured in meters.

A distinction is made between the standard deviation σ (sigma) of a whole population or of a random variable, and the standard deviation s of a subset-population sample. The formulae are given below.

The term standard deviation was introduced to statistics by Karl Pearson (On the Dissection of Asymmetrical Frequency Curves, 1894).


Dark blue is less than one standard deviation from the mean. For the normal distribution, this accounts for 68.26% of the set. For the normal distribution, two standard deviations from the mean (blue and brown) account for 95.46%. For the normal distribution, three standard deviations (blue, brown and green) account for 99.73%.