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The **chi-squared distribution **with *k* degrees of freedom is the distribution of a sum of the squares of *k* independent, standard normal random variables. It’s called a chi *squared* because it describes the summation of squares of the normally distributed random variables.

The degrees of freedom (*k*) are equal to the number of samples being summed. For example, if you have taken 11 samples from the normal distribution, then *df* = 11. The degrees of freedom (11 in this example) are equal to the **mean **of the chi-squared distribution.

The probability density function is:

Where

*k*= degrees of freedom- Γ (
*k*/2) = the gamma function.

Chi square distributions are always right skewed. As the degrees of freedom increase, the chi-squared distribution will look more and more like a normal distribution.

The cumulative density function (CDF) is given by

## Related Distributions

The chi-squared distribution is a special case of the gamma distribution. A chi-squared distribution with *k *degrees of freedom is equal to a gamma distribution with *a* = *k*/ 2 and *b* = 0.5 (or β = 2).

A chi-bar-squared distribution is a mixture of chi-square distributions, mixed over their degrees of freedom.

## Uses of The Chi-Squared Distribution

The chi-squared distribution has a variety of different uses, including [1]:

- Confidence interval estimation for a population standard deviation of a normal distribution from a sample standard deviation [2].
- Independence of two criteria of classification of qualitative variables.
- Relationships between categorical variables (contingency tables).
- Sample variance study when the underlying distribution is normal.
- Tests of deviations of differences between expected and observed frequencies (one-way tables).
- The chi-square goodness of fit test.

**References**

[1] Stephanie Glen. “Chi-Square Statistic: How to Calculate It / Distribution” From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/probability-and-statistics/chi-square/

[2] Johns Hopkins. http://ocw.jhsph.edu/courses/fundepiii/PDFs/Lecture17.pdf

## One response to “Chi-Squared Distribution”

[…] justifying the “skew-normal” name. For example, if X has a Azzalini PDF, then X2 follows a chi-squared distribution with one degree of freedom for all values of X. In applied statistics, the distribution can be used […]