# 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.

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 |