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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
( Xi - X )2 / N ]
where σ is the population standard deviation,
X
is the population mean, Xi is the ith 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 ( xi
- x )2 / ( n - 1 ) ]
where s is the sample standard deviation, x is
the sample mean, xi is the ith 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.