# Statistics Dictionary

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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:

Χ2 = [ ( n - 1 ) * s2 ] / σ2

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 = Y0 * ( Χ2 ) ( v/2 - 1 ) * e - Χ2 / 2

where Y0 is a constant that depends on the number of degrees of freedom, Χ2 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). Y0 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).