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# Statistics Dictionary

To see a definition, select a term from the dropdown text box below. The statistics dictionary will display the definition, plus links to related web pages.

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### Chi-Square Distribution

Suppose we conduct the following
statistical experiment
. We select a random sample of size *n* from
a normal population, having a standard deviation equal to σ.
We find that the standard deviation in our sample is equal to *s*. Given
these data, we can compute a statistic, called **chi-square**,
using the following equation:

Χ^{2} =
[ ( n - 1 ) * s^{2} ] / σ^{2}

The distribution of the chi-square statistic is called the chi-square distribution. The **
chi-square
distribution
** is defined by the following
probability density function
:

Y = Y_{0} * ( Χ^{2}
) ^{( v/2 - 1 )} * *e*^{
- Χ2
/ 2
}

where Y_{0} is a constant that depends on the number of degrees of
freedom, Χ^{2} is
the chi-square statistic, *v* = *n* - 1 is the number of
degrees of freedom
, and *e* is a constant equal to the base of the
natural logarithm system (approximately 2.71828). Y_{0} is defined, so
that the area under the chi-square curve is equal to one.

In the figure above, the red curve shows the distribution of chi-square values
computed from all possible samples of size 3, where degrees of freedom is *n*
- 1 = 3 - 1 = 2. Similarly, the green curve shows the distribution for
samples of size 5 (degrees of freedom equal to 4); and the blue curve, for
samples of size 11 (degrees of freedom equal to 10).

See also: | Tutorial: Chi-Square Probability Distribution | Chi-Square Calculator |