# Standard deviation

A distribution of values can be characterised in several ways, but most commonly by:

• a measure of the central tendency of the collection of values (eg mean, median, mode)
• a measure of the spread or dispersion of values around the central tendency

The standard deviation is the most commonly used measure of the dispersion of values. The collection of values makes up a probability distribution.

In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is defined as the square root of the variance.

The standard deviation is measured in the same units as the values of the population. For a population of distances in meters, the standard deviation is also measured in meters, whereas the variance is measured in square meters.

At least 75% of the values in any population are at most two standard deviations away from the mean (see Chebyshev's inequality). The actual percentage depends on the distribution, for example it is approximately 95% if the population has a normal distribution .

The term standard deviation was introduced to statistics by Karl Pearson in On the dissection of asymmetrical frequency curves, 1894 .

The standard deviation is the root mean square (RMS) deviation of the values from their arithmetic mean . For example, in the population {4, 8}, the mean is 6 and the standard deviation is 2. This may be written: {4, 8} ≈ 6±2. In this case 100% of the values in the population are within one standard deviation of the mean.

Standard deviation is the most common measure of statistical dispersion, measuring how widely spread the values in a data set are. If the data points are all close to the mean, then the standard deviation is close to zero. If many data points are far from the mean, then the standard deviation is far from zero. If all the data values are equal, then the standard deviation is zero.

The standard deviation σ of a statistical population can be estimated by a modified standard deviation (s) of a statistical sample.