# The Chi-square distribution

Example:
Prob( X^2 >= | Degrees of Freedom= ) <=

X^2 =
>
Degrees of Freedom =
>

Characteristics:
This is not a test, but a distribution. The Chi-square distribution, is derived from the > Normal distribution. It is the distribution of a sum of squared Normal distributed variables. That is, if all Xi are independent and all have an identical, standard Normal distribution then X^2 = X1*X1 + X2*X2 + X3*X3 + ... + Xv*Xv is Chi-square distributed with v degrees of freedom with mean = v and variance = 2*v.
The importance of the Chi-square distribution stems from the fact that it describes the distribution of the Variance of a sample taken from a Normal distributed population.
Note that X^2 is a Sum of Squares and should be tested One-Sided only.

H0:
The distribution of the underlying sample values X has mean = 0 and variance = 1, i.e., is Standard Normal. As a consequence, X^2 has mean equal to the Degrees of Freedom and a variance equal to 2 * Degrees of Freedom.

Assumptions:
The values from which X^2 is calculated are themselves Normal distributed with unit variance.

Scale:
Interval

Procedure:
-

Level of significance:
Use a table to look up the level of significance associated with X^2 and the Degrees of Freedom.

Approximation:
If the Degrees of Freedom > 30, the distribution of

Z = {(X^2/DoF)^(1/3) - (1 - 2/(9*DoF))}/SQRT(2/(9*DoF))

can be approximated by a > Standard Normal Distribution.

Remarks:
The Chi-Square distribution is based on a sum of squares, therefore the value of X^2 will always be larger than (or equal to) zero. This means that all testing should be done One-Tailed only. This is in fact done in the above example.
WARNING: the level of significance given here is only an approximation, take care when using it! (use a table if necessary)