Editing Chi-squared distribution (section)

=== Asymptotic properties ===
[[File:Chi-square median approx.png|thumb|upright=1.818|Approximate formula for median (from the Wilson–Hilferty transformation) compared with numerical quantile (top); and difference ({{font color|blue|blue}}) and relative difference ({{font color|red|red}}) between numerical quantile and approximate formula (bottom). For the chi-squared distribution, only the positive integer numbers of degrees of freedom (circles) are meaningful.]]

By the [[central limit theorem]], because the chi-squared distribution is the sum of <math>k</math> independent random variables with finite mean and variance, it converges to a normal distribution for large <math>k</math>. For many practical purposes, for <math>k>50</math> the distribution is sufficiently close to a [[normal distribution]], so the difference is ignorable.<ref>{{cite book|title=Statistics for experimenters|author=Box, Hunter and Hunter|publisher=Wiley|year=1978|isbn=978-0-471-09315-2|page=[https://archive.org/details/statisticsforexp00geor/page/118 118]|url-access=registration|url=https://archive.org/details/statisticsforexp00geor/page/118}}</ref> Specifically, if <math>X \sim \chi^2(k)</math>, then as <math>k</math> tends to infinity, the distribution of <math>(X-k)/\sqrt{2k}</math> [[convergence of random variables#Convergence in distribution|tends]] to a standard normal distribution. However, convergence is slow as the [[skewness]] is <math>\sqrt{8/k}</math> and the [[excess kurtosis]] is <math>12/k</math>.

The sampling distribution of <math>\ln(\chi^2)</math> converges to normality much faster than the sampling distribution of <math>\chi^2</math>,<ref>{{cite journal |first1=M. S. |last1=Bartlett |first2=D. G. |last2=Kendall |title=The Statistical Analysis of Variance-Heterogeneity and the Logarithmic Transformation |journal=Supplement to the Journal of the Royal Statistical Society |volume=8 |issue=1 |year=1946 |pages=128–138 |jstor=2983618 |doi=10.2307/2983618 }}</ref> as the [[logarithmic transformation|logarithmic transform]] removes much of the asymmetry.<ref name="Pillai-2016">{{Cite journal|last=Pillai|first=Natesh S.|year=2016|title=An unexpected encounter with Cauchy and Lévy|journal=[[Annals of Statistics]]|volume=44|issue=5|pages=2089–2097|doi=10.1214/15-aos1407|arxiv=1505.01957|s2cid=31582370}}</ref>

Other functions of the chi-squared distribution converge more rapidly to a normal distribution. Some examples are:
* If <math>X \sim \chi^2(k)</math> then <math>\sqrt{2X}</math> is approximately normally distributed with mean <math>\sqrt{2k-1}</math> and unit variance (1922, by [[R. A. Fisher]], see (18.23), p.&nbsp;426 of Johnson.<ref name="Johnson-1994" />
* If <math>X \sim \chi^2(k)</math> then <math>\sqrt[3]{X/k}</math> is approximately normally distributed with mean <math> 1-\frac{2}{9k}</math> and variance <math>\frac{2}{9k} .</math><ref>{{cite journal |last1=Wilson |first1=E. B. |last2=Hilferty |first2=M. M. |year=1931 |title=The distribution of chi-squared |journal=[[Proc. Natl. Acad. Sci. USA]] |volume=17 |issue=12 |pages=684–688 |bibcode=1931PNAS...17..684W |doi=10.1073/pnas.17.12.684 |pmid=16577411 |pmc=1076144 |doi-access=free }}</ref> This is known as the '''Wilson–Hilferty transformation''', see (18.24), p.&nbsp;426 of Johnson.<ref name="Johnson-1994" />
** This [[Data transformation (statistics)#Transforming to normality|normalizing transformation]] leads directly to the commonly used median approximation <math>k\bigg(1-\frac{2}{9k}\bigg)^3\;</math> by back-transforming from the mean, which is also the median, of the normal distribution.