# Statistics and Probability Dictionary

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### Standard Deviation

The standard deviation is a numerical value used to indicate how widely individuals in a group vary. If individual observations vary greatly from the group mean, the standard deviation is big; and vice versa.

It is important to distinguish between the standard deviation of a population
and the standard deviation of a sample. They have different notation, and they
are computed differently. The standard deviation of a population is denoted by
σ and the standard deviation of a sample, by *s*.

The standard deviation of a population is defined by the following formula:

σ = sqrt [ Σ
( X_{i} - X )^{2} / N ]

where σ is the population standard deviation,
X
is the population mean, X_{i} is the *i*th element
from the population, and N is the number of elements in the population.

The standard deviation of a sample is defined by slightly different formula:

*s* = sqrt [ Σ ( x_{i}
- x )^{2} / ( n - 1 ) ]

where *s* is the sample standard deviation, x is
the sample mean, x_{i} is the *i*th element from the sample, and n
is the number of elements in the sample. Using this equation, the standard deviation
of the sample is an unbiased estimate of the standard deviation of the population.

And finally, the standard deviation is equal to the square root of the variance.

See also: | Statistics Tutorial: Measures of Variability | AP Statistics Tutorial: Measures of Variability | Random Variable Attributes |