Chi-Squared Distribution

< Back to Probability Distribution List

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:


  • k = degrees of freedom
  • Γ (k/2) = the gamma function.
PDF for the chi-squared distribution.

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 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.


[1] Stephanie Glen. “Chi-Square Statistic: How to Calculate It / Distribution” From Elementary Statistics for the rest of us!

[2] Johns Hopkins.

One response to “Chi-Squared Distribution”