This is a chi-squared distribution with one degree of freedom.

Typically the square of the difference is compared to a chi-squared distribution.

This is a special case of the generalized chi-squared distribution.

This is an important intuition for understanding the chi-squared distribution.

Generate having a chi-squared distribution with 2P + 2 degrees of freedom.

Then T has a chi-squared distribution with n 1 degrees of freedom.

The following are proofs of several characteristics related to the chi-squared distribution.

To derive the chi-squared distribution with 2 degrees of freedom, there could be several methods.

The distribution of X is a chi-squared distribution for the following reason.

One formulation of the generalized chi-squared distribution is as follows.

The resulting value can be compared to the chi-squared distribution to determine the goodness of fit.

This has approximately a chi-squared distribution with k 1 df.

Moreover, tables for the normal, Student's t and chi-squared distributions are given as well.

It can therefore be regarded as a generalized chi-squared distribution, for even degrees of freedom.

The approximation to the chi-squared distribution breaks down if expected frequencies are too low.

Following are some of the most common situations in which the chi-squared distribution arises from a Gaussian-distributed sample.

A chi-squared distribution shows many low bids but few high bids.

Then should be rejected with significance if the value is greater than the critical value of the appropriate chi-squared distribution.

Further properties of the chi-squared distribution can be found in the box at the upper right corner of this article.

"A modified score test statistic having chi-squared distribution to order ", Biometrika, 1991.

The estimator s will be proportional to the chi-squared distribution:

But if n is small, then the probabilities based on chi-squared distributions may not be very close approximations.

If estimates with no bias, then the noncentrality is zero and follows a central chi-squared distribution.

This is again a scaled inverse chi-squared distribution, with parameters and .

The square of a standard normal random variable has a chi-squared distribution with one degree of freedom.