Editing Chi-squared distribution (section)

== Occurrence and applications{{anchor|Applications}} ==
The chi-squared distribution has numerous applications in inferential [[statistics]], for instance in [[chi-squared test]]s and in estimating [[variance]]s. It enters the problem of estimating the mean of a normally distributed population and the problem of estimating the slope of a [[linear regression|regression]] line via its role in [[Student's t-distribution]]. It enters all [[analysis of variance]] problems via its role in the [[F-distribution]], which is the distribution of the ratio of two independent chi-squared [[random variable]]s, each divided by their respective degrees of freedom.

Following are some of the most common situations in which the chi-squared distribution arises from a Gaussian-distributed sample.
* if <math>X_1, ..., X_n</math> are [[i.i.d.]] <math>N(\mu, \sigma^2)</math> [[random variable]]s, then <math>\sum_{i=1}^n(X_i - \overline{X})^2 \sim \sigma^2 \chi^2_{n-1}</math> where <math>\overline{X} = \frac{1}{n} \sum_{i=1}^n X_i</math>.
* The box below shows some [[statistics]] based on <math>X_i \sim N(\mu_i, \sigma^2_i), i= 1, \ldots, k</math> independent random variables that have probability distributions related to the chi-squared distribution:
{| class="wikitable"  style="margin:1em auto;" align="center"
|-
! Name !! Statistic
|-
| chi-squared distribution || <math>\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2</math>
|-
| [[noncentral chi-squared distribution]] || <math>\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2</math>
|-
| [[chi distribution]] || <math>\sqrt{\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2}</math>
|-
| [[noncentral chi distribution]] || <math>\sqrt{\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2}</math>
|}
The chi-squared distribution is also often encountered in [[magnetic resonance imaging]].<ref>den Dekker A. J., Sijbers J., (2014) "Data distributions in magnetic resonance images: a review", ''Physica Medica'', [https://dx.doi.org/10.1016/j.ejmp.2014.05.002]</ref>