chi-square distribution

• カイ二乗分布、χ二乗分布

関

chi-square test

WordNet

1. raise to the second power
2. (geometry) a plane rectangle with four equal sides and four right angles; a four-sided regular polygon; "you can compute the area of a square if you know the length of its sides" (同)foursquare
3. the product of two equal terms; "nine is the second power of three"; "gravity is inversely proportional to the square of the distance" (同)second power
4. make square; "Square the circle"; "square the wood with a file" (同)square_up
5. something approximating the shape of a square
6. a hand tool consisting of two straight arms at right angles; used to construct or test right angles; "the carpenter who built this room must have lost his square"
7. any artifact having a shape similar to a plane geometric figure with four equal sides and four right angles; "a checkerboard has 64 squares"
8. someone who doesnt understand what is going on (同)lame
9. a formal and conservative person with old-fashioned views (同)square toes
10. rigidly conventional or old-fashioned (同)straight
11. without evasion or compromise; "a square contradiction"; "he is not being as straightforward as it appears" (同)straightforward, straight
12. be compatible with; "one idea squares with another"
13. cause to match, as of ideas or acts
14. having four equal sides and four right angles or forming a right angle; "a square peg in a round hole"; "a square corner"
15. leaving no balance; "my account with you is now all square"
16. pay someone and settle a debt; "I squared with him"
17. position so as to be square; "He squared his shoulders"
18. the spatial or geographic property of being scattered about over a range, area, or volume; "worldwide in distribution"; "the distribution of nerve fibers"; "in complementary distribution" (同)dispersion
19. the act of distributing or spreading or apportioning
20. the commercial activity of transporting and selling goods from a producer to a consumer
21. (statistics) an arrangement of values of a variable showing their observed or theoretical frequency of occurrence (同)statistical_distribution
22. the 22nd letter of the Greek alphabet (同)khi
23. having been made square

PrepTutorEJDIC

1. 『正方形』;四角な物;(チェス・チェッカーなどの盤の,正方形の)目,ます目 / (四角い)『広場』(街路の交差点にあって,しばしば中央に植木や芝などが植えてあり,小公園になっている);《おもに英》広場の回りの建物(街路)(《略》Sq.) / (四方を街路で囲まれた方形の)一区画,ブロック / 直角定規,かね尺 / (数の)『2乗』,平方(《略》sq.) / 《俗》旧式な人 / 『正方形の』,四角な,直角の,直角をなす / 角ばった,がっかりした / 『平方の』,2乗の(《略》『sq.』) / 《補語にのみ用いて》対等の,五分五分の(even);貸し借りにない / 正直な(honest),公正な(fair),正しい(just) / 率直な,はっきりした,きっぱりした(direct) / 実質のある,十分な / 《俗》しゃちほこばった / =squarely / …‘を'正方形(四角)にする;直角にする / …‘を'正方形(四角)に区切る《+『off』+『名』,+『名』+『off』》 / 〈肩・ひじなど〉‘を'張る / (人と)〈勘定〉‘を'決済する,清算する《+『名』〈勘定〉+『with』+『名』〈人〉》 / (…と)…‘を'一致させる,適合させる《+『名』+『with』+『名』》 / 《受動態で》〈数〉‘を'2乗する;〈ある形・図形など〉‘の'平方積(面積)を求める / 〈人〉‘を'買収する,抱き込む,…‘に'わいろを使う / 〈試合の得点〉‘を'同点にする / (…と)一致する,適合する《+『with』+『名』》
2. 〈U〉(…への)『配分』,『分配』,配布,配達《+『to』+『名』》 / 〈U〉分布;分布区域 / 〈C〉配分されたもの,配給品 / 〈U〉(商品の)流通[組織,機構]
3. キー(ギリシア語アルファベットの第22字X,x;英語のchに相当)

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2017/05/26 23:36:20」(JST)

wiki en

[Wiki en表示]

This article is about the mathematics of the chi-squared distribution. For its uses in statistics, see chi-squared test. For the music group, see Chi2 (band).

chi-squared

Probability density function
Cumulative distribution function
Notation	$$ χ 2 ( k ) {\displaystyle \chi ^{2}(k)\!} or $$ χ k 2 {\displaystyle \chi _{k}^{2}\!}
Parameters	$$ k ∈ N > 0     {\displaystyle k\in \mathbb {N} _{>0}~~} (known as "degrees of freedom")
Support	$$ x ∈ [ 0 , + ∞ ) {\displaystyle x\in [0,+\infty )}
PDF	$$ 1 2 k 2 Γ ( k 2 ) x k 2 − 1 e − x 2 {\displaystyle {\frac {1}{2^{\frac {k}{2}}\Gamma \left({\frac {k}{2}}\right)}}\;x^{{\frac {k}{2}}-1}e^{-{\frac {x}{2}}}\,}
CDF	$$ 1 − 1 Γ ( k 2 ) γ ( k 2 , x 2 ) {\displaystyle 1-{\frac {1}{\Gamma \left({\frac {k}{2}}\right)}}\;\gamma \left({\tfrac {k}{2}},\,{\frac {x}{2}}\right)}
Mean	$$ k {\displaystyle k}
Median	$$ ≈ k ( 1 − 2 9 k ) 3 {\displaystyle \approx k{\bigg (}1-{\frac {2}{9k}}{\bigg )}^{3}}
Mode	$$ max ( k − 2 , 0 ) {\displaystyle \max(k-2,0)}
Variance	$$ 2 k {\displaystyle 2k}
Skewness	$$ 8 / k {\displaystyle \scriptstyle {\sqrt {8/k}}\,}
Ex. kurtosis	$$ 12 k {\displaystyle {\frac {12}{k}}}
Entropy	$$ k 2 + ln ⁡ ( 2 Γ ( k 2 ) ) + ( 1 − k 2 ) ψ ( k 2 ) (nats) {\displaystyle {\begin{aligned}{\tfrac {k}{2}}&+\ln(2\Gamma ({\tfrac {k}{2}}))\\&\!+(1-{\tfrac {k}{2}})\psi ({\tfrac {k}{2}})\,{\scriptstyle {\text{(nats)}}}\end{aligned}}}
MGF	$$ ( 1 − 2 t ) − k 2  for  t < 1 2 {\displaystyle (1-2t)^{-{\frac {k}{2}}}{\text{ for }}t<{\frac {1}{2}}}
CF	$$ ( 1 − 2 i t ) − k 2 {\displaystyle (1-2it)^{-{\frac {k}{2}}}}

In probability theory and statistics, the chi-squared distribution (also chi-square or χ²-distribution) with $k$ degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. It is a special case of the gamma distribution and is one of the most widely used probability distributions in inferential statistics, e. g., in hypothesis testing or in construction of confidence intervals. When it is being distinguished from the more general noncentral chi-squared distribution, this distribution is sometimes called the central chi-squared distribution.

The chi-squared distribution is used in the common chi-squared tests for goodness of fit of an observed distribution to a theoretical one, the independence of two criteria of classification of qualitative data, and in confidence interval estimation for a population standard deviation of a normal distribution from a sample standard deviation. Many other statistical tests also use this distribution, like Friedman's analysis of variance by ranks.

Definition

If Z₁, ..., Z_k are independent, standard normal random variables, then the sum of their squares,

$Q\ =\sum _{i=1}^{k}Z_{i}^{2},$

is distributed according to the chi-squared distribution with k degrees of freedom. This is usually denoted as

$Q\ \sim \ \chi ^{2}(k)\ \ {\text{or}}\ \ Q\ \sim \ \chi _{k}^{2}.$

The chi-squared distribution has one parameter: k — a positive integer that specifies the number of degrees of freedom (i. e. the number of Z_i’s)

Introduction

The chi-squared distribution is used primarily in hypothesis testing. Unlike more widely known distributions such as the normal distribution and the exponential distribution, the chi-squared distribution is rarely used to model natural phenomena. It arises in the following hypothesis tests, among others.

• Chi-squared test of independence in contingency tables
• Chi-squared test of goodness of fit of observed data to hypothetical distributions
• Likelihood-ratio test for nested models
• Log-rank test in survival analysis
• Cochran–Mantel–Haenszel test for stratified contingency tables

It is also a component of the definition of the t-distribution and the F-distribution used in t-tests, analysis of variance, and regression analysis.

The primary reason that the chi-squared distribution is used extensively in hypothesis testing is its relationship to the normal distribution. Many hypothesis tests use a test statistic, such as the t statistic in a t-test. For these hypothesis tests, as the sample size, n, increases, the sampling distribution of the test statistic approaches the normal distribution (Central Limit Theorem). Because the test statistic (such as t) is asymptotically normally distributed, provided the sample size is sufficiently large, the distribution used for hypothesis testing may be approximated by a normal distribution. Testing hypotheses using a normal distribution is well understood and relatively easy. The simplest chi-squared distribution is the square of a standard normal distribution. So wherever a normal distribution could be used for a hypothesis test, a chi-squared distribution could be used.

Specifically, suppose that Z is a standard normal random variable, with mean = 0 and variance = 1. Z ~ N(0,1). A sample drawn at random from Z is a sample from the distribution shown in the graph of the standard normal distribution. Define a new random variable Q. To generate a random sample from Q, take a sample from Z and square the value. The distribution of the squared values is given by the random variable Q = Z². The distribution of the random variable Q is an example of a chi-squared distribution: $\ Q\ \sim \ \chi _{1}^{2}.$ The subscript 1 indicates that this particular chi-squared distribution is constructed from only 1 standard normal distribution. A chi-squared distribution constructed by squaring a single standard normal distribution is said to have 1 degree of freedom. Thus, as the sample size for a hypothesis test increases, the distribution of the test statistic approaches a normal distribution, and the distribution of the square of the test statistic approaches a chi-squared distribution. Just as extreme values of the normal distribution have low probability (and give small p-values), extreme values of the chi-squared distribution have low probability.

An additional reason that the chi-squared distribution is widely used is that it is a member of the class of likelihood ratio tests (LRT). LRT's have several desirable properties; in particular, LRT's commonly provide the highest power to reject the null hypothesis (Neyman–Pearson lemma). However, the normal and chi-squared approximations are only valid asymptotically. For this reason, it is preferable to use the t distribution rather than the normal approximation or the chi-squared approximation for small sample size. Similarly, in analyses of contingency tables, the chi-squared approximation will be poor for small sample size, and it is preferable to use Fisher's exact test. Ramsey and Ramsey show that the exact binomial test is always more powerful than the normal approximation.

Lancaster shows the connections among the binomial, normal, and chi-squared distributions, as follows. De Moivre and Laplace established that a binomial distribution could be approximated by a normal distribution. Specifically they showed the asymptotic normality of the random variable

$\chi ={m-Np \over {\sqrt {(Npq)}}}$

where m is the observed number of successes in N trials, where the probability of success is p, and q = 1 − p.

Squaring both sides of the equation gives

$\chi ^{2}={(m-Np)^{2} \over (Npq)}$

Using N = Np + N(1 − p), N = m + (N − m), and q = 1 − p, this equation simplifies to

$\chi ^{2}={(m-Np)^{2} \over (Np)}+{(N-m-Nq)^{2} \over (Nq)}$

The expression on the right is of the form that Pearson would generalize to the form:

$\chi ^{2}=\sum _{i=1}^{n}{\frac {(O_{i}-E_{i})^{2}}{E_{i}}}$

where

$\chi ^{2}$ = Pearson's cumulative test statistic, which asymptotically approaches a $\chi ^{2}$ distribution.

$O_{i}$ = the number of observations of type i.

$E_{i}=Np_{i}$ = the expected (theoretical) frequency of type i, asserted by the null hypothesis that the fraction of type i in the population is $p_{i}$

$n$ = the number of cells in the table.

In the case of a binomial outcome (flipping a coin), the binomial distribution may be approximated by a normal distribution (for sufficiently large n). Because the square of a standard normal distribution is the chi-squared distribution with one degree of freedom, the probability of a result such as 1 heads in 10 trials can be approximated either by the normal or the chi-squared distribution. However, many problems involve more than the two possible outcomes of a binomial, and instead require 3 or more categories, which leads to the multinomial distribution. Just as de Moivre and Laplace sought for and found the normal approximation to the binomial, Pearson sought for and found a multivariate normal approximation to the multinomial distribution. Pearson showed that the chi-squared distribution, the sum of multiple normal distributions, was such an approximation to the multinomial distribution

Characteristics

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

Probability density function

The probability density function (pdf) of the chi-square distribution is

${\displaystyle f(x;\,k)={\begin{cases}{\dfrac {x^{(k/2-1)}e^{-x/2}}{2^{k/2}\Gamma \left({\frac {k}{2}}\right)}},&x>0;\ ==Wikipedia preview== ==Wikipedia preview==$出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2017/07/09 02:54:29」(JST) ====[http://en.wikipedia.org/wiki/chi-square%20distribution wiki en]====

This article is about the mathematics of the chi-squared distribution. For its uses in statistics, see chi-squared test. For the music group, see Chi2 (band).

chi-squared

Probability density function
Cumulative distribution function
Notation	$$ χ 2 ( k ) {\displaystyle \chi ^{2}(k)\!} or $$ χ k 2 {\displaystyle \chi _{k}^{2}\!}
Parameters	$$ k ∈ N > 0     {\displaystyle k\in \mathbb {N} _{>0}~~} (known as "degrees of freedom")
Support	$$ x ∈ [ 0 , + ∞ ) {\displaystyle x\in [0,+\infty )}
PDF	$$ 1 2 k 2 Γ ( k 2 ) x k 2 − 1 e − x 2 {\displaystyle {\frac {1}{2^{\frac {k}{2}}\Gamma \left({\frac {k}{2}}\right)}}\;x^{{\frac {k}{2}}-1}e^{-{\frac {x}{2}}}\,}
CDF	$$ 1 Γ ( k 2 ) γ ( k 2 , x 2 ) {\displaystyle {\frac {1}{\Gamma \left({\frac {k}{2}}\right)}}\;\gamma \left({\tfrac {k}{2}},\,{\frac {x}{2}}\right)}
Mean	$$ k {\displaystyle k}
Median	$$ ≈ k ( 1 − 2 9 k ) 3 {\displaystyle \approx k{\bigg (}1-{\frac {2}{9k}}{\bigg )}^{3}}
Mode	$$ max ( k − 2 , 0 ) {\displaystyle \max(k-2,0)}
Variance	$$ 2 k {\displaystyle 2k}
Skewness	$$ 8 / k {\displaystyle \scriptstyle {\sqrt {8/k}}\,}
Ex. kurtosis	$$ 12 k {\displaystyle {\frac {12}{k}}}
Entropy	$$ k 2 + ln ⁡ ( 2 Γ ( k 2 ) ) + ( 1 − k 2 ) ψ ( k 2 ) (nats) {\displaystyle {\begin{aligned}{\tfrac {k}{2}}&+\ln(2\Gamma ({\tfrac {k}{2}}))\\&\!+(1-{\tfrac {k}{2}})\psi ({\tfrac {k}{2}})\,{\scriptstyle {\text{(nats)}}}\end{aligned}}}
MGF	$$ ( 1 − 2 t ) − k 2  for  t < 1 2 {\displaystyle (1-2t)^{-{\frac {k}{2}}}{\text{ for }}t<{\frac {1}{2}}}
CF	$$ ( 1 − 2 i t ) − k 2 {\displaystyle (1-2it)^{-{\frac {k}{2}}}}

In probability theory and statistics, the chi-squared distribution (also chi-square or χ²-distribution) with $k$ degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-square distribution is a special case of the gamma distribution and is one of the most widely used probability distributions in inferential statistics, e. g., in hypothesis testing or in construction of confidence intervals. When it is being distinguished from the more general noncentral chi-squared distribution, this distribution is sometimes called the central chi-squared distribution.

The chi-squared distribution is used in the common chi-squared tests for goodness of fit of an observed distribution to a theoretical one, the independence of two criteria of classification of qualitative data, and in confidence interval estimation for a population standard deviation of a normal distribution from a sample standard deviation. Many other statistical tests also use this distribution, such as Friedman's analysis of variance by ranks.

Definition

If Z₁, ..., Z_k are independent, standard normal random variables, then the sum of their squares,

$Q\ =\sum _{i=1}^{k}Z_{i}^{2},$

is distributed according to the chi-squared distribution with k degrees of freedom. This is usually denoted as

$Q\ \sim \ \chi ^{2}(k)\ \ {\text{or}}\ \ Q\ \sim \ \chi _{k}^{2}.$

The chi-squared distribution has one parameter: k — a positive integer that specifies the number of degrees of freedom (i. e. the number of Z_i’s)

Introduction

The chi-squared distribution is used primarily in hypothesis testing. Unlike more widely known distributions such as the normal distribution and the exponential distribution, the chi-squared distribution is rarely used to model natural phenomena. It arises in the following hypothesis tests, among others.

• Chi-squared test of independence in contingency tables
• Chi-squared test of goodness of fit of observed data to hypothetical distributions
• Likelihood-ratio test for nested models
• Log-rank test in survival analysis
• Cochran–Mantel–Haenszel test for stratified contingency tables

It is also a component of the definition of the t-distribution and the F-distribution used in t-tests, analysis of variance, and regression analysis.

The primary reason that the chi-squared distribution is used extensively in hypothesis testing is its relationship to the normal distribution. Many hypothesis tests use a test statistic, such as the t statistic in a t-test. For these hypothesis tests, as the sample size, n, increases, the sampling distribution of the test statistic approaches the normal distribution (Central Limit Theorem). Because the test statistic (such as t) is asymptotically normally distributed, provided the sample size is sufficiently large, the distribution used for hypothesis testing may be approximated by a normal distribution. Testing hypotheses using a normal distribution is well understood and relatively easy. The simplest chi-squared distribution is the square of a standard normal distribution. So wherever a normal distribution could be used for a hypothesis test, a chi-squared distribution could be used.

Specifically, suppose that Z is a standard normal random variable, with mean = 0 and variance = 1. Z ~ N(0,1). A sample drawn at random from Z is a sample from the distribution shown in the graph of the standard normal distribution. Define a new random variable Q. To generate a random sample from Q, take a sample from Z and square the value. The distribution of the squared values is given by the random variable Q = Z². The distribution of the random variable Q is an example of a chi-squared distribution: $\ Q\ \sim \ \chi _{1}^{2}.$ The subscript 1 indicates that this particular chi-squared distribution is constructed from only 1 standard normal distribution. A chi-squared distribution constructed by squaring a single standard normal distribution is said to have 1 degree of freedom. Thus, as the sample size for a hypothesis test increases, the distribution of the test statistic approaches a normal distribution, and the distribution of the square of the test statistic approaches a chi-squared distribution. Just as extreme values of the normal distribution have low probability (and give small p-values), extreme values of the chi-squared distribution have low probability.

An additional reason that the chi-squared distribution is widely used is that it is a member of the class of likelihood ratio tests (LRT). LRT's have several desirable properties; in particular, LRT's commonly provide the highest power to reject the null hypothesis (Neyman–Pearson lemma). However, the normal and chi-squared approximations are only valid asymptotically. For this reason, it is preferable to use the t distribution rather than the normal approximation or the chi-squared approximation for small sample size. Similarly, in analyses of contingency tables, the chi-squared approximation will be poor for small sample size, and it is preferable to use Fisher's exact test. Ramsey and Ramsey show that the exact binomial test is always more powerful than the normal approximation.

Lancaster shows the connections among the binomial, normal, and chi-squared distributions, as follows. De Moivre and Laplace established that a binomial distribution could be approximated by a normal distribution. Specifically they showed the asymptotic normality of the random variable

$\chi ={m-Np \over {\sqrt {(Npq)}}}$

where m is the observed number of successes in N trials, where the probability of success is p, and q = 1 − p.

Squaring both sides of the equation gives

$\chi ^{2}={(m-Np)^{2} \over (Npq)}$

Using N = Np + N(1 − p), N = m + (N − m), and q = 1 − p, this equation simplifies to

$\chi ^{2}={(m-Np)^{2} \over (Np)}+{(N-m-Nq)^{2} \over (Nq)}$

The expression on the right is of the form that Pearson would generalize to the form:

$\chi ^{2}=\sum _{i=1}^{n}{\frac {(O_{i}-E_{i})^{2}}{E_{i}}}$

where

$\chi ^{2}$ = Pearson's cumulative test statistic, which asymptotically approaches a $\chi ^{2}$ distribution.

$O_{i}$ = the number of observations of type i.

$E_{i}=Np_{i}$ = the expected (theoretical) frequency of type i, asserted by the null hypothesis that the fraction of type i in the population is $p_{i}$

$n$ = the number of cells in the table.

In the case of a binomial outcome (flipping a coin), the binomial distribution may be approximated by a normal distribution (for sufficiently large n). Because the square of a standard normal distribution is the chi-squared distribution with one degree of freedom, the probability of a result such as 1 heads in 10 trials can be approximated either by the normal or the chi-squared distribution. However, many problems involve more than the two possible outcomes of a binomial, and instead require 3 or more categories, which leads to the multinomial distribution. Just as de Moivre and Laplace sought for and found the normal approximation to the binomial, Pearson sought for and found a multivariate normal approximation to the multinomial distribution. Pearson showed that the chi-squared distribution, the sum of multiple normal distributions, was such an approximation to the multinomial distribution

Characteristics

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

Probability density function

The probability density function (pdf) of the chi-square distribution is

${\displaystyle f(x;\,k)={\begin{cases}{\dfrac {x^{(k/2-1)}e^{-x/2}}{2^{k/2}\Gamma \left({\frac {k}{2}}\right)}},&x>0;\ ==Wikipedia preview== ==Wikipedia preview==$出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2017/09/23 11:39:39」(JST) ====[http://en.wikipedia.org/wiki/chi-square%20distribution wiki en]====

This article is about the mathematics of the chi-squared distribution. For its uses in statistics, see chi-squared test. For the music group, see Chi2 (band).

chi-squared

Probability density function
Cumulative distribution function
Notation	$$ χ 2 ( k ) {\displaystyle \chi ^{2}(k)\!} or $$ χ k 2 {\displaystyle \chi _{k}^{2}\!}
Parameters	$$ k ∈ N > 0     {\displaystyle k\in \mathbb {N} _{>0}~~} (known as "degrees of freedom")
Support	$$ x ∈ [ 0 , + ∞ ) {\displaystyle x\in [0,+\infty )}
PDF	$$ 1 2 k 2 Γ ( k 2 ) x k 2 − 1 e − x 2 {\displaystyle {\frac {1}{2^{\frac {k}{2}}\Gamma \left({\frac {k}{2}}\right)}}\;x^{{\frac {k}{2}}-1}e^{-{\frac {x}{2}}}\,}
CDF	$$ 1 Γ ( k 2 ) γ ( k 2 , x 2 ) {\displaystyle {\frac {1}{\Gamma \left({\frac {k}{2}}\right)}}\;\gamma \left({\frac {k}{2}},\,{\frac {x}{2}}\right)}
Mean	$$ k {\displaystyle k}
Median	$$ ≈ k ( 1 − 2 9 k ) 3 {\displaystyle \approx k{\bigg (}1-{\frac {2}{9k}}{\bigg )}^{3}}
Mode	$$ max ( k − 2 , 0 ) {\displaystyle \max(k-2,0)}
Variance	$$ 2 k {\displaystyle 2k}
Skewness	$$ 8 / k {\displaystyle \scriptstyle {\sqrt {8/k}}\,}
Ex. kurtosis	$$ 12 k {\displaystyle {\frac {12}{k}}}
Entropy	$$ k 2 + ln ⁡ ( 2 Γ ( k 2 ) ) + ( 1 − k 2 ) ψ ( k 2 ) (nats) {\displaystyle {\begin{aligned}{\tfrac {k}{2}}&+\ln(2\Gamma ({\tfrac {k}{2}}))\\&\!+(1-{\tfrac {k}{2}})\psi ({\tfrac {k}{2}})\,{\scriptstyle {\text{(nats)}}}\end{aligned}}}
MGF	$$ ( 1 − 2 t ) − k 2  for  t < 1 2 {\displaystyle (1-2t)^{-{\frac {k}{2}}}{\text{ for }}t<{\frac {1}{2}}}
CF	$$ ( 1 − 2 i t ) − k 2 {\displaystyle (1-2it)^{-{\frac {k}{2}}}}

In probability theory and statistics, the chi-squared distribution (also chi-square or χ²-distribution) with $k$ degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-square distribution is a special case of the gamma distribution and is one of the most widely used probability distributions in inferential statistics, e. g., in hypothesis testing or in construction of confidence intervals. When it is being distinguished from the more general noncentral chi-squared distribution, this distribution is sometimes called the central chi-squared distribution.

The chi-squared distribution is used in the common chi-squared tests for goodness of fit of an observed distribution to a theoretical one, the independence of two criteria of classification of qualitative data, and in confidence interval estimation for a population standard deviation of a normal distribution from a sample standard deviation. Many other statistical tests also use this distribution, such as Friedman's analysis of variance by ranks.

Definition

If Z₁, ..., Z_k are independent, standard normal random variables, then the sum of their squares,

$Q\ =\sum _{i=1}^{k}Z_{i}^{2},$

is distributed according to the chi-squared distribution with k degrees of freedom. This is usually denoted as

$Q\ \sim \ \chi ^{2}(k)\ \ {\text{or}}\ \ Q\ \sim \ \chi _{k}^{2}.$

The chi-squared distribution has one parameter: k — a positive integer that specifies the number of degrees of freedom (i. e. the number of Z_i’s)

Introduction

The chi-squared distribution is used primarily in hypothesis testing. Unlike more widely known distributions such as the normal distribution and the exponential distribution, the chi-squared distribution is rarely used to model natural phenomena. It arises in the following hypothesis tests, among others.

• Chi-squared test of independence in contingency tables
• Chi-squared test of goodness of fit of observed data to hypothetical distributions
• Likelihood-ratio test for nested models
• Log-rank test in survival analysis
• Cochran–Mantel–Haenszel test for stratified contingency tables

It is also a component of the definition of the t-distribution and the F-distribution used in t-tests, analysis of variance, and regression analysis.

The primary reason that the chi-squared distribution is used extensively in hypothesis testing is its relationship to the normal distribution. Many hypothesis tests use a test statistic, such as the t statistic in a t-test. For these hypothesis tests, as the sample size, n, increases, the sampling distribution of the test statistic approaches the normal distribution (Central Limit Theorem). Because the test statistic (such as t) is asymptotically normally distributed, provided the sample size is sufficiently large, the distribution used for hypothesis testing may be approximated by a normal distribution. Testing hypotheses using a normal distribution is well understood and relatively easy. The simplest chi-squared distribution is the square of a standard normal distribution. So wherever a normal distribution could be used for a hypothesis test, a chi-squared distribution could be used.

Specifically, suppose that Z is a standard normal random variable, with mean = 0 and variance = 1. Z ~ N(0,1). A sample drawn at random from Z is a sample from the distribution shown in the graph of the standard normal distribution. Define a new random variable Q. To generate a random sample from Q, take a sample from Z and square the value. The distribution of the squared values is given by the random variable Q = Z². The distribution of the random variable Q is an example of a chi-squared distribution: $\ Q\ \sim \ \chi _{1}^{2}.$ The subscript 1 indicates that this particular chi-squared distribution is constructed from only 1 standard normal distribution. A chi-squared distribution constructed by squaring a single standard normal distribution is said to have 1 degree of freedom. Thus, as the sample size for a hypothesis test increases, the distribution of the test statistic approaches a normal distribution, and the distribution of the square of the test statistic approaches a chi-squared distribution. Just as extreme values of the normal distribution have low probability (and give small p-values), extreme values of the chi-squared distribution have low probability.

An additional reason that the chi-squared distribution is widely used is that it is a member of the class of likelihood ratio tests (LRT). LRT's have several desirable properties; in particular, LRT's commonly provide the highest power to reject the null hypothesis (Neyman–Pearson lemma). However, the normal and chi-squared approximations are only valid asymptotically. For this reason, it is preferable to use the t distribution rather than the normal approximation or the chi-squared approximation for small sample size. Similarly, in analyses of contingency tables, the chi-squared approximation will be poor for small sample size, and it is preferable to use Fisher's exact test. Ramsey and Ramsey show that the exact binomial test is always more powerful than the normal approximation.

Lancaster shows the connections among the binomial, normal, and chi-squared distributions, as follows. De Moivre and Laplace established that a binomial distribution could be approximated by a normal distribution. Specifically they showed the asymptotic normality of the random variable

$\chi ={m-Np \over {\sqrt {(Npq)}}}$

where m is the observed number of successes in N trials, where the probability of success is p, and q = 1 − p.

Squaring both sides of the equation gives

$\chi ^{2}={(m-Np)^{2} \over (Npq)}$

Using N = Np + N(1 − p), N = m + (N − m), and q = 1 − p, this equation simplifies to

$\chi ^{2}={(m-Np)^{2} \over (Np)}+{(N-m-Nq)^{2} \over (Nq)}$

The expression on the right is of the form that Pearson would generalize to the form:

$\chi ^{2}=\sum _{i=1}^{n}{\frac {(O_{i}-E_{i})^{2}}{E_{i}}}$

where

$\chi ^{2}$ = Pearson's cumulative test statistic, which asymptotically approaches a $\chi ^{2}$ distribution.

$O_{i}$ = the number of observations of type i.

$E_{i}=Np_{i}$ = the expected (theoretical) frequency of type i, asserted by the null hypothesis that the fraction of type i in the population is $p_{i}$

$n$ = the number of cells in the table.

In the case of a binomial outcome (flipping a coin), the binomial distribution may be approximated by a normal distribution (for sufficiently large n). Because the square of a standard normal distribution is the chi-squared distribution with one degree of freedom, the probability of a result such as 1 heads in 10 trials can be approximated either by the normal or the chi-squared distribution. However, many problems involve more than the two possible outcomes of a binomial, and instead require 3 or more categories, which leads to the multinomial distribution. Just as de Moivre and Laplace sought for and found the normal approximation to the binomial, Pearson sought for and found a multivariate normal approximation to the multinomial distribution. Pearson showed that the chi-squared distribution, the sum of multiple normal distributions, was such an approximation to the multinomial distribution

Characteristics

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

Probability density function

The probability density function (pdf) of the chi-square distribution is

${\displaystyle f(x;\,k)={\begin{cases}{\dfrac {x^{(k/2-1)}e^{-x/2}}{2^{k/2}\Gamma \left({\frac {k}{2}}\right)}},&x>0;\ ==Wikipedia preview== ==Wikipedia preview==$出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2018/10/01 05:18:00」(JST) ====[http://en.wikipedia.org/wiki/chi-square%20distribution wiki en]====

This article is about the mathematics of the chi-squared distribution. For its uses in statistics, see chi-squared test. For the music group, see Chi2 (band).

chi-squared

Probability density function
Cumulative distribution function
Notation	$$ χ 2 ( k ) {\displaystyle \chi ^{2}(k)\;}

or $$

 
   
     
       
         χ
         
           k
         
         
           2
         
       
       
     
   
   {\displaystyle \chi _{k}^{2}\!}
 

Parameters

$$

 
   
     
       k
       ∈
       
         
           N
         
         
           >
           0
         
       
        
        
     
   
   {\displaystyle k\in \mathbb {N} _{>0}~~}
 

(known as "degrees of freedom")Support

$$

 
   
     
       x
       ∈
       (
       0
       ,
       +
       ∞
       )
       
     
   
   {\displaystyle x\in (0,+\infty )\;}
 

if $$

 
   
     
       k
       =
       1
     
   
   {\displaystyle k=1}
 

, otherwise $$

 
   
     
       x
       ∈
       [
       0
       ,
       +
       ∞
       )
       
     
   
   {\displaystyle x\in [0,+\infty )\;}
 

PDF

$$

 
   
     
       
         
           1
           
             
               2
               
                 k
                 
                   /
                 
                 2
               
             
             Γ
             (
             k
             
               /
             
             2
             )
           
         
       
       
       
         x
         
           k
           
             /
           
           2
           −
           1
         
       
       
         e
         
           −
           x
           
             /
           
           2
         
       
       
     
   
   {\displaystyle {\frac {1}{2^{k/2}\Gamma (k/2)}}\;x^{k/2-1}e^{-x/2}\;}
 

CDF

$$

 
   
     
       
         
           1
           
             Γ
             (
             k
             
               /
             
             2
             )
           
         
       
       
       γ
       
         (
         
           
             
               k
               2
             
           
           ,
           
           
             
               x
               2
             
           
         
         )
       
       
     
   
   {\displaystyle {\frac {1}{\Gamma (k/2)}}\;\gamma \left({\frac {k}{2}},\,{\frac {x}{2}}\right)\;}
 

Mean

$$

 
   
     
       k
     
   
   {\displaystyle k}
 

Median

$$

 
   
     
       ≈
       k
       
         
           (
         
       
       1
       −
       
         
           2
           
             9
             k
           
         
       
       
         
           
             )
           
         
         
           3
         
       
       
     
   
   {\displaystyle \approx k{\bigg (}1-{\frac {2}{9k}}{\bigg )}^{3}\;}
 

Mode

$$

 
   
     
       max
       (
       k
       −
       2
       ,
       0
       )
       
     
   
   {\displaystyle \max(k-2,0)\;}
 

Variance

$$

 
   
     
       2
       k
       
     
   
   {\displaystyle 2k\;}
 

Skewness

$$

 
   
     
       
         
           
             8
             
               /
             
             k
           
         
         
       
     
   
   {\displaystyle \scriptstyle {\sqrt {8/k}}\,}
 

Ex. kurtosis

$$

 
   
     
       
         
           12
           k
         
       
     
   
   {\displaystyle {\frac {12}{k}}}
 

Entropy

 
   
     
       
         
           
             
               
                 
                   
                     k
                     2
                   
                 
               
             
             
               
               +
               log
               ⁡
               (
               2
               Γ
               (
               k
               
                 /
               
               2
               )
               )
             
           
           
             
             
               
               
               +
               (
               1
               −
               k
               
                 /
               
               2
               )
               ψ
               (
               k
               
                 /
               
               2
               )
               
               
                 
                   
                     (nats)
                   
                 
               
             
           
         
       
     
   
   {\displaystyle {\begin{aligned}{\tfrac {k}{2}}&+\log(2\Gamma (k/2))\\&\!+(1-k/2)\psi (k/2)\,{\scriptstyle {\text{(nats)}}}\end{aligned}}}
 

MGF

 
   
     
       (
       1
       −
       2
       t
       
         )
         
           −
           k
           
             /
           
           2
         
       
       
          for 
       
       t
       <
       
         
           1
           2
         
       
       
     
   
   {\displaystyle (1-2t)^{-k/2}{\text{ for }}t<{\frac {1}{2}}\;}
 

CF

$$

 
   
     
       (
       1
       −
       2
       i
       t
       
         )
         
           −
           k
           
             /
           
           2
         
       
     
   
   {\displaystyle (1-2it)^{-k/2}}
 

      PGF

 
   
     
       (
       1
       −
       2
       ln
       ⁡
       t
       
         )
         
           −
           k
           
             /
           
           2
         
       
       
          for 
       
       0
       <
       t
       <
       
         
           e
         
       
       
     
   
   {\displaystyle (1-2\ln t)^{-k/2}{\text{ for }}0<t<{\sqrt {e}}\;}
 

In probability theory and statistics, the chi-squared distribution (also chi-square or χ²-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-square distribution is a special case of the gamma distribution and is one of the most widely used probability distributions in inferential statistics, notably in hypothesis testing or in construction of confidence intervals. When it is being distinguished from the more general noncentral chi-squared distribution, this distribution is sometimes called the central chi-squared distribution.

The chi-squared distribution is used in the common chi-squared tests for goodness of fit of an observed distribution to a theoretical one, the independence of two criteria of classification of qualitative data, and in confidence interval estimation for a population standard deviation of a normal distribution from a sample standard deviation. Many other statistical tests also use this distribution, such as Friedman's analysis of variance by ranks.

Definition

If Z₁, ..., Z_k are independent, standard normal random variables, then the sum of their squares,

$Q\ =\sum _{i=1}^{k}Z_{i}^{2},$

is distributed according to the chi-squared distribution with k degrees of freedom. This is usually denoted as

$Q\ \sim \ \chi ^{2}(k)\ \ {\text{or}}\ \ Q\ \sim \ \chi _{k}^{2}.$

The chi-squared distribution has one parameter: k — a positive integer that specifies the number of degrees of freedom (the number of Z_i’s).

Introduction

The chi-squared distribution is used primarily in hypothesis testing. Unlike more widely known distributions such as the normal distribution and the exponential distribution, the chi-squared distribution is not as often applied in the direct modeling of natural phenomena. It arises in the following hypothesis tests, among others.

• Chi-squared test of independence in contingency tables
• Chi-squared test of goodness of fit of observed data to hypothetical distributions
• Likelihood-ratio test for nested models
• Log-rank test in survival analysis
• Cochran–Mantel–Haenszel test for stratified contingency tables

It is also a component of the definition of the t-distribution and the F-distribution used in t-tests, analysis of variance, and regression analysis.

The primary reason that the chi-squared distribution is used extensively in hypothesis testing is its relationship to the normal distribution. Many hypothesis tests use a test statistic, such as the t-statistic in a t-test. For these hypothesis tests, as the sample size, n, increases, the sampling distribution of the test statistic approaches the normal distribution (central limit theorem). Because the test statistic (such as t) is asymptotically normally distributed, provided the sample size is sufficiently large, the distribution used for hypothesis testing may be approximated by a normal distribution. Testing hypotheses using a normal distribution is well understood and relatively easy. The simplest chi-squared distribution is the square of a standard normal distribution. So wherever a normal distribution could be used for a hypothesis test, a chi-squared distribution could be used.

Specifically, suppose that Z is a standard normal random variable, with mean = 0 and variance = 1. Z ~ N(0,1). A sample drawn at random from Z is a sample from the distribution shown in the graph of the standard normal distribution.

Define a new random variable Q. To generate a random sample from Q, take a sample from Z and square the value. The distribution of the squared values is given by the random variable Q = Z². The distribution of the random variable Q is an example of a chi-squared distribution: $$

 
   
     
        
       Q
        
       ∼
        
       
         χ
         
           1
         
         
           2
         
       
       .
     
   
   {\displaystyle \ Q\ \sim \ \chi _{1}^{2}.}
 

The subscript 1 indicates that this particular chi-squared distribution is constructed from only 1 standard normal distribution. A chi-squared distribution constructed by squaring a single standard normal distribution is said to have 1 degree of freedom. Thus, as the sample size for a hypothesis test increases, the distribution of the test statistic approaches a normal distribution, and the distribution of the square of the test statistic approaches a chi-squared distribution. Just as extreme values of the normal distribution have low probability (and give small p-values), extreme values of the chi-squared distribution have low probability.

An additional reason that the chi-squared distribution is widely used is that it is a member of the class of likelihood ratio tests (LRT). LRT's have several desirable properties; in particular, LRT's commonly provide the highest power to reject the null hypothesis (Neyman–Pearson lemma). However, the normal and chi-squared approximations are only valid asymptotically. For this reason, it is preferable to use the t distribution rather than the normal approximation or the chi-squared approximation for small sample size. Similarly, in analyses of contingency tables, the chi-squared approximation will be poor for small sample size, and it is preferable to use Fisher's exact test. Ramsey shows that the exact binomial test is always more powerful than the normal approximation.

Lancaster

shows the connections among the binomial, normal, and chi-squared distributions, as follows. De Moivre and Laplace established that a binomial distribution could be approximated by a normal distribution. Specifically they showed the asymptotic normality of the random variable

$\chi ={m-Np \over {\sqrt {Npq}}}$

where m is the observed number of successes in N trials, where the probability of success is p, and q = 1 − p.

Squaring both sides of the equation gives

$\chi ^{2}={(m-Np)^{2} \over Npq}$

Using N = Np + N(1 − p), N = m + (N − m), and q = 1 − p, this equation simplifies to

$\chi ^{2}={(m-Np)^{2} \over Np}+{(N-m-Nq)^{2} \over Nq}$

The expression on the right is of the form that Pearson would generalize to the form:

$\chi ^{2}=\sum _{i=1}^{n}{\frac {(O_{i}-E_{i})^{2}}{E_{i}}}$

where

$\chi ^{2}$ = Pearson's cumulative test statistic, which asymptotically approaches a $\chi ^{2}$ distribution.

$O_{i}$ = the number of observations of type i.

$E_{i}=Np_{i}$ = the expected (theoretical) frequency of type i, asserted by the null hypothesis that the fraction of type i in the population is $p_{i}$

$n$ = the number of cells in the table.

In the case of a binomial outcome (flipping a coin), the binomial distribution may be approximated by a normal distribution (for sufficiently large n). Because the square of a standard normal distribution is the chi-squared distribution with one degree of freedom, the probability of a result such as 1 heads in 10 trials can be approximated either by the normal or the chi-squared distribution. However, many problems involve more than the two possible outcomes of a binomial, and instead require 3 or more categories, which leads to the multinomial distribution. Just as de Moivre and Laplace sought for and found the normal approximation to the binomial, Pearson sought for and found a multivariate normal approximation to the multinomial distribution. Pearson showed that the chi-squared distribution, the sum of multiple normal distributions, was such an approximation to the multinomial distribution

Characteristics

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

Probability density function

The probability density function (pdf) of the chi-square distribution is

====[http://en.wikipedia.org/wiki/chi-square%20distribution wiki en]====

This article is about the mathematics of the chi-squared distribution. For its uses in statistics, see chi-squared test. For the music group, see Chi2 (band).

chi-squared

Probability density function
Cumulative distribution function
Notation	$$ χ 2 ( k ) {\displaystyle \chi ^{2}(k)\;} or $$ χ k 2 {\displaystyle \chi _{k}^{2}\!}
Parameters	$$ k ∈ N > 0     {\displaystyle k\in \mathbb {N} _{>0}~~} (known as "degrees of freedom")
Support	$$ x ∈ ( 0 , + ∞ ) {\displaystyle x\in (0,+\infty )\;} if $$ k = 1 {\displaystyle k=1} , otherwise $$ x ∈ [ 0 , + ∞ ) {\displaystyle x\in [0,+\infty )\;}
PDF	$$ 1 2 k / 2 Γ ( k / 2 ) x k / 2 − 1 e − x / 2 {\displaystyle {\frac {1}{2^{k/2}\Gamma (k/2)}}\;x^{k/2-1}e^{-x/2}\;}
CDF	$$ 1 Γ ( k / 2 ) γ ( k 2 , x 2 ) {\displaystyle {\frac {1}{\Gamma (k/2)}}\;\gamma \left({\frac {k}{2}},\,{\frac {x}{2}}\right)\;}
Mean	$$ k {\displaystyle k}
Median	$$ ≈ k ( 1 − 2 9 k ) 3 {\displaystyle \approx k{\bigg (}1-{\frac {2}{9k}}{\bigg )}^{3}\;}
Mode	$$ max ( k − 2 , 0 ) {\displaystyle \max(k-2,0)\;}
Variance	$$ 2 k {\displaystyle 2k\;}
Skewness	$$ 8 / k {\displaystyle \scriptstyle {\sqrt {8/k}}\,}
Ex. kurtosis	$$ 12 k {\displaystyle {\frac {12}{k}}}
Entropy	k 2 + log ⁡ ( 2 Γ ( k / 2 ) ) + ( 1 − k / 2 ) ψ ( k / 2 ) (nats) {\displaystyle {\begin{aligned}{\tfrac {k}{2}}&+\log(2\Gamma (k/2))\\&\!+(1-k/2)\psi (k/2)\,{\scriptstyle {\text{(nats)}}}\end{aligned}}}
MGF	( 1 − 2 t ) − k / 2  for  t < 1 2 {\displaystyle (1-2t)^{-k/2}{\text{ for }}t<{\frac {1}{2}}\;}
CF	$$ ( 1 − 2 i t ) − k / 2 {\displaystyle (1-2it)^{-k/2}}
PGF	( 1 − 2 ln ⁡ t ) − k / 2  for  0 < t < e {\displaystyle (1-2\ln t)^{-k/2}{\text{ for }}0<t<{\sqrt {e}}\;}

In probability theory and statistics, the chi-squared distribution (also chi-square or χ²-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-square distribution is a special case of the gamma distribution and is one of the most widely used probability distributions in inferential statistics, notably in hypothesis testing or in construction of confidence intervals. When it is being distinguished from the more general noncentral chi-squared distribution, this distribution is sometimes called the central chi-squared distribution.

The chi-squared distribution is used in the common chi-squared tests for goodness of fit of an observed distribution to a theoretical one, the independence of two criteria of classification of qualitative data, and in confidence interval estimation for a population standard deviation of a normal distribution from a sample standard deviation. Many other statistical tests also use this distribution, such as Friedman's analysis of variance by ranks.

Definition

If Z₁, ..., Z_k are independent, standard normal random variables, then the sum of their squares,

$Q\ =\sum _{i=1}^{k}Z_{i}^{2},$

is distributed according to the chi-squared distribution with k degrees of freedom. This is usually denoted as

$Q\ \sim \ \chi ^{2}(k)\ \ {\text{or}}\ \ Q\ \sim \ \chi _{k}^{2}.$

The chi-squared distribution has one parameter: k, a positive integer that specifies the number of degrees of freedom (the number of Z_i’s).

Introduction

The chi-squared distribution is used primarily in hypothesis testing. Unlike more widely known distributions such as the normal distribution and the exponential distribution, the chi-squared distribution is not as often applied in the direct modeling of natural phenomena. It arises in the following hypothesis tests, among others.

• Chi-squared test of independence in contingency tables
• Chi-squared test of goodness of fit of observed data to hypothetical distributions
• Likelihood-ratio test for nested models
• Log-rank test in survival analysis
• Cochran–Mantel–Haenszel test for stratified contingency tables

It is also a component of the definition of the t-distribution and the F-distribution used in t-tests, analysis of variance, and regression analysis.

The primary reason that the chi-squared distribution is used extensively in hypothesis testing is its relationship to the normal distribution. Many hypothesis tests use a test statistic, such as the t-statistic in a t-test. For these hypothesis tests, as the sample size, n, increases, the sampling distribution of the test statistic approaches the normal distribution (central limit theorem). Because the test statistic (such as t) is asymptotically normally distributed, provided the sample size is sufficiently large, the distribution used for hypothesis testing may be approximated by a normal distribution. Testing hypotheses using a normal distribution is well understood and relatively easy. The simplest chi-squared distribution is the square of a standard normal distribution. So wherever a normal distribution could be used for a hypothesis test, a chi-squared distribution could be used.

Specifically, suppose that Z is a standard normal random variable, with mean = 0 and variance = 1. Z ~ N(0,1). A sample drawn at random from Z is a sample from the distribution shown in the graph of the standard normal distribution.

Define a new random variable Q. To generate a random sample from Q, take a sample from Z and square the value. The distribution of the squared values is given by the random variable Q = Z². The distribution of the random variable Q is an example of a chi-squared distribution: $$

 
   
     
        
       Q
        
       ∼
        
       
         χ
         
           1
         
         
           2
         
       
       .
     
   
   {\displaystyle \ Q\ \sim \ \chi _{1}^{2}.}
 

The subscript 1 indicates that this particular chi-squared distribution is constructed from only 1 standard normal distribution. A chi-squared distribution constructed by squaring a single standard normal distribution is said to have 1 degree of freedom. Thus, as the sample size for a hypothesis test increases, the distribution of the test statistic approaches a normal distribution, and the distribution of the square of the test statistic approaches a chi-squared distribution. Just as extreme values of the normal distribution have low probability (and give small p-values), extreme values of the chi-squared distribution have low probability.

An additional reason that the chi-squared distribution is widely used is that it is a member of the class of likelihood ratio tests (LRT). LRT's have several desirable properties; in particular, LRT's commonly provide the highest power to reject the null hypothesis (Neyman–Pearson lemma). However, the normal and chi-squared approximations are only valid asymptotically. For this reason, it is preferable to use the t distribution rather than the normal approximation or the chi-squared approximation for small sample size. Similarly, in analyses of contingency tables, the chi-squared approximation will be poor for small sample size, and it is preferable to use Fisher's exact test. Ramsey shows that the exact binomial test is always more powerful than the normal approximation.

Lancaster shows the connections among the binomial, normal, and chi-squared distributions, as follows. De Moivre and Laplace established that a binomial distribution could be approximated by a normal distribution. Specifically they showed the asymptotic normality of the random variable

$\chi ={m-Np \over {\sqrt {Npq}}}$

where m is the observed number of successes in N trials, where the probability of success is p, and q = 1 − p.

Squaring both sides of the equation gives

$\chi ^{2}={(m-Np)^{2} \over Npq}$

Using N = Np + N(1 − p), N = m + (N − m), and q = 1 − p, this equation simplifies to

$\chi ^{2}={(m-Np)^{2} \over Np}+{(N-m-Nq)^{2} \over Nq}$

The expression on the right is of the form that Pearson would generalize to the form:

$\chi ^{2}=\sum _{i=1}^{n}{\frac {(O_{i}-E_{i})^{2}}{E_{i}}}$

where

$\chi ^{2}$ = Pearson's cumulative test statistic, which asymptotically approaches a $\chi ^{2}$ distribution.

$O_{i}$ = the number of observations of type i.

$E_{i}=Np_{i}$ = the expected (theoretical) frequency of type i, asserted by the null hypothesis that the fraction of type i in the population is $p_{i}$

$n$ = the number of cells in the table.

In the case of a binomial outcome (flipping a coin), the binomial distribution may be approximated by a normal distribution (for sufficiently large n). Because the square of a standard normal distribution is the chi-squared distribution with one degree of freedom, the probability of a result such as 1 heads in 10 trials can be approximated either by the normal or the chi-squared distribution. However, many problems involve more than the two possible outcomes of a binomial, and instead require 3 or more categories, which leads to the multinomial distribution. Just as de Moivre and Laplace sought for and found the normal approximation to the binomial, Pearson sought for and found a multivariate normal approximation to the multinomial distribution. Pearson showed that the chi-squared distribution, the sum of multiple normal distributions, was such an approximation to the multinomial distribution

Characteristics

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

Probability density function

The probability density function (pdf) of the chi-square distribution is

$f(x;\,k)={\begin{cases}{\dfrac {x^{{\frac {k}{2}}-1}e^{-{\frac {x}{2}}}}{2^{\frac {k}{2}}\Gamma \left({\frac {k}{2}}\right)}},&x>0;\,&{\text{otherwise}}.\end{cases}},&{\text{otherwise}}.\end{cases}}}$

where ${\textstyle \Gamma (k/2)}$ denotes the gamma function, which has closed-form values for integer k.

For derivations of the pdf in the cases of one, two and k degrees of freedom, see Proofs related to chi-squared distribution.

Cumulative distribution function

Chernoff bound for the CDF and tail (1-CDF) of a chi-squared random variable with ten degrees of freedom (k = 10)

Its cumulative distribution function is:

$F(x;\,k)={\frac {\gamma ({\frac {k}{2}},\,{\frac {x}{2}})}{\Gamma ({\frac {k}{2}})}}=P\left({\frac {k}{2}},\,{\frac {x}{2}}\right),$

where $\gamma (s,t)$ is the lower incomplete gamma function and ${\textstyle P(s,t)}$ is the regularized gamma function.

In a special case of k = 2 this function has a simple form:

$F(x;\,2)=1-e^{-x/2}$

and the integer recurrence of the gamma function makes it easy to compute for other small even k.

Tables of the chi-squared cumulative distribution function are widely available and the function is included in many spreadsheets and all statistical packages.

Letting $$

 
   
     
       z
       ≡
       x
       
         /
       
       k
     
   
   {\displaystyle z\equiv x/k}
 

, Chernoff bounds on the lower and upper tails of the CDF may be obtained. For the cases when

 
   
     
       0
       <
       z
       <
       1
     
   
   {\displaystyle 0<z<1}
 

(which include all of the cases when this CDF is less than half):

$F(zk;\,k)\leq (ze^{1-z})^{k/2}.$

The tail bound for the cases when $z>1$, similarly, is

$1-F(zk;\,k)\leq (ze^{1-z})^{k/2}.$

For another approximation for the CDF modeled after the cube of a Gaussian, see under Noncentral chi-squared distribution.

Additivity

It follows from the definition of the chi-squared distribution that the sum of independent chi-squared variables is also chi-squared distributed. Specifically, if {X_i}_i=1ⁿ are independent chi-squared variables with {k_i}_i=1ⁿ degrees of freedom, respectively, then Y = X₁ + ⋯ + X_n is chi-squared distributed with k₁ + ⋯ + k_n degrees of freedom.

Sample mean

The sample mean of $n$ i.i.d. chi-squared variables of degree $k$ is distributed according to a gamma distribution with shape $\alpha$ and scale $\theta$ parameters:

${\bar {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}\sim \operatorname {Gamma} \left(\alpha =n\,k/2,\theta =2/n\right)\qquad {\text{where }}X_{i}\sim \chi ^{2}(k)$

Asymptotically, given that for a scale parameter $\alpha$ going to infinity, a Gamma distribution converges towards a normal distribution with expectation $\mu =\alpha \cdot \theta$ and variance $\sigma ^{2}=\alpha \,\theta ^{2}$, the sample mean converges towards:

${\bar {X}}{\xrightarrow {n\to \infty }}N(\mu =k,\sigma ^{2}=2\,k/n)$

Note that we would have obtained the same result invoking instead the central limit theorem, noting that for each chi-squared variable of degree $k$ the expectation is $k$ , and its variance $2\,k$ (and hence the variance of the sample mean ${\bar {X}}$ being $\sigma ^{2}=2\,k/n$).

Entropy

The differential entropy is given by

$h=\int _{0}^{\infty }f(x;\,k)\ln f(x;\,k)\,dx={\frac {k}{2}}+\ln \left[2\,\Gamma \left({\frac {k}{2}}\right)\right]+\left(1-{\frac {k}{2}}\right)\,\psi \!\left[{\frac {k}{2}}\right],$

where ψ(x) is the Digamma function.

The chi-squared distribution is the maximum entropy probability distribution for a random variate X for which $$

 
   
     
       E
       ⁡
       (
       X
       )
       =
       k
     
   
   {\displaystyle \operatorname {E} (X)=k}
 

and $$

 
   
     
       E
       ⁡
       (
       ln
       ⁡
       (
       X
       )
       )
       =
       ψ
       (
       k
       
         /
       
       2
       )
       +
       ln
       ⁡
       (
       2
       )
     
   
   {\displaystyle \operatorname {E} (\ln(X))=\psi (k/2)+\ln(2)}
 

are fixed. Since the chi-squared is in the family of gamma distributions, this can be derived by substituting appropriate values in the Expectation of the log moment of gamma. For derivation from more basic principles, see the derivation in moment-generating function of the sufficient statistic.

Noncentral moments

The moments about zero of a chi-squared distribution with k degrees of freedom are given by

$\operatorname {E} (X^{m})=k(k+2)(k+4)\cdots (k+2m-2)=2^{m}{\frac {\Gamma \left(m+{\frac {k}{2}}\right)}{\Gamma \left({\frac {k}{2}}\right)}}.$

Cumulants

The cumulants are readily obtained by a (formal) power series expansion of the logarithm of the characteristic function:

$\kappa _{n}=2^{n-1}(n-1)!\,k$

Asymptotic properties

By the central limit theorem, because the chi-squared distribution is the sum of k independent random variables with finite mean and variance, it converges to a normal distribution for large k. For many practical purposes, for k > 50 the distribution is sufficiently close to a normal distribution for the difference to be ignored. Specifically, if X ~ χ²(k), then as k tends to infinity, the distribution of $(X-k)/{\sqrt {2k}}$ tends to a standard normal distribution. However, convergence is slow as the skewness is ${\sqrt {8/k}}$ and the excess kurtosis is 12/k.

The sampling distribution of ln(χ²) converges to normality much faster than the sampling distribution of χ², as the logarithm removes much of the asymmetry. Other functions of the chi-squared distribution converge more rapidly to a normal distribution. Some examples are:

• If X ~ χ²(k) then $\scriptstyle {\sqrt {2X}}$ is approximately normally distributed with mean $\scriptstyle {\sqrt {2k-1}}$ and unit variance (1922, by R. A. Fisher, see (18.23), p. 426 of.
• If X ~ χ²(k) then $\scriptstyle {\sqrt[{3}]{X/k}}$ is approximately normally distributed with mean $\scriptstyle 1-2/(9k)$ and variance $\scriptstyle 2/(9k).$ This is known as the Wilson–Hilferty transformation, see (18.24), p. 426 of.

Relation to other distributions

Approximate formula for median compared with numerical quantile (top). Difference between numerical quantile and approximate formula (bottom).

• As $k\to \infty$, $(\chi _{k}^{2}-k)/{\sqrt {2k}}~{\xrightarrow {d}}\ N(0,1)\,$ (normal distribution)
• $\chi _{k}^{2}\sim {\chi '}_{k}^{2}(0)$ (noncentral chi-squared distribution with non-centrality parameter $\lambda =0$)
• If $Y\sim \mathrm {F} (\nu _{1},\nu _{2})$ then $X=\lim _{\nu _{2}\to \infty }\nu _{1}Y$ has the chi-squared distribution $\chi _{\nu _{1}}^{2}$

• As a special case, if $Y\sim \mathrm {F} (1,\nu _{2})\,$ then $X=\lim _{\nu _{2}\to \infty }Y\,$ has the chi-squared distribution $\chi _{1}^{2}$

• $\|{\boldsymbol {N}}_{i=1,\ldots ,k}(0,1)\|^{2}\sim \chi _{k}^{2}$ (The squared norm of k standard normally distributed variables is a chi-squared distribution with k degrees of freedom)
• If $X\sim \chi ^{2}(\nu )\,$ and $c>0\,$, then $cX\sim \Gamma (k=\nu /2,\theta =2c)\,$. (gamma distribution)
• If $X\sim \chi _{k}^{2}$ then ${\sqrt {X}}\sim \chi _{k}$ (chi distribution)
• If $X\sim \chi ^{2}(2)$, then $X\sim \operatorname {Exp} (1/2)$ is an exponential distribution. (See gamma distribution for more.)
• If $X\sim \operatorname {Rayleigh} (1)\,$ (Rayleigh distribution) then $X^{2}\sim \chi ^{2}(2)\,$
• If $X\sim \operatorname {Maxwell} (1)\,$ (Maxwell distribution) then $X^{2}\sim \chi ^{2}(3)\,$
• If $X\sim \chi ^{2}(\nu )$ then ${\tfrac {1}{X}}\sim \operatorname {Inv-} \chi ^{2}(\nu )\,$ (Inverse-chi-squared distribution)
• The chi-squared distribution is a special case of type 3 Pearson distribution
• If $X\sim \chi ^{2}(\nu _{1})\,$ and $Y\sim \chi ^{2}(\nu _{2})\,$ are independent then ${\tfrac {X}{X+Y}}\sim \operatorname {Beta} ({\tfrac {\nu _{1}}{2}},{\tfrac {\nu _{2}}{2}})\,$ (beta distribution)
• If $X\sim \operatorname {U} (0,1)\,$ (uniform distribution) then $-2\log(X)\sim \chi ^{2}(2)\,$
• $\chi ^{2}(6)\,$ is a transformation of Laplace distribution
• If $X_{i}\sim \operatorname {Laplace} (\mu ,\beta )\,$ then $\sum _{i=1}^{n}{\frac {2|X_{i}-\mu |}{\beta }}\sim \chi ^{2}(2n)\,$
• If $X_{i}$ follows the generalized normal distribution (version 1) with parameters $\mu ,\alpha ,\beta$ then $\sum _{i=1}^{n}{\frac {2|X_{i}-\mu |^{\beta }}{\alpha }}\sim \chi ^{2}\left({\frac {2n}{\beta }}\right)\,$
• chi-squared distribution is a transformation of Pareto distribution
• Student's t-distribution is a transformation of chi-squared distribution
• Student's t-distribution can be obtained from chi-squared distribution and normal distribution
• Noncentral beta distribution can be obtained as a transformation of chi-squared distribution and Noncentral chi-squared distribution
• Noncentral t-distribution can be obtained from normal distribution and chi-squared distribution

A chi-squared variable with k degrees of freedom is defined as the sum of the squares of k independent standard normal random variables.

If Y is a k-dimensional Gaussian random vector with mean vector μ and rank k covariance matrix C, then X = (Y−μ)^TC⁻¹(Y − μ) is chi-squared distributed with k degrees of freedom.

The sum of squares of statistically independent unit-variance Gaussian variables which do not have mean zero yields a generalization of the chi-squared distribution called the noncentral chi-squared distribution.

If Y is a vector of k i.i.d. standard normal random variables and A is a k×k symmetric, idempotent matrix with rank k−n then the quadratic form Y^TAY is chi-squared distributed with k−n degrees of freedom.

If $$

 
   
     
       Σ
     
   
   {\displaystyle \Sigma }
 

is a $$

 
   
     
       p
       ×
       p
     
   
   {\displaystyle p\times p}
 

positive-semidefinite covariance matrix with strictly positive diagonal entries, then for $$

 
   
     
       X
       ∼
       N
       (
       0
       ,
       Σ
       )
     
   
   {\displaystyle X\sim N(0,\Sigma )}
 

and $$

 
   
     
       w
     
   
   {\displaystyle w}
 

a random $$

 
   
     
       p
     
   
   {\displaystyle p}
 

-vector independent of $$

 
   
     
       X
     
   
   {\displaystyle X}
 

such that $$

 
   
     
       
         w
         
           1
         
       
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       ⋯
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       =
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   {\displaystyle w_{1}+\cdots +w_{p}=1}
 

and $$

 
   
     
       
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   {\displaystyle w_{i}\geq 0,i=1,\cdots ,p,}
 

it holds that

$$

 
   
     
       
         
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The chi-squared distribution is also naturally related to other distributions arising from the Gaussian. In particular,

• Y is F-distributed, Y ~ F(k₁,k₂) if $\scriptstyle Y={\frac {X_{1}/k_{1}}{X_{2}/k_{2}}}$ where X₁ ~ χ²(k₁) and X₂  ~ χ²(k₂) are statistically independent.
• If X₁  ~  χ²_k1 and X₂  ~  χ²_k2 are statistically independent, then X₁ + X₂  ~ χ²_k1+k2. If X₁ and X₂ are not independent, then X₁ + X₂ is not chi-squared distributed.

Generalizations

The chi-squared distribution is obtained as the sum of the squares of k independent, zero-mean, unit-variance Gaussian random variables. Generalizations of this distribution can be obtained by summing the squares of other types of Gaussian random variables. Several such distributions are described below.

Linear combination

If $X_{1},\ldots ,X_{n}$ are chi square random variables and $a_{1},\ldots ,a_{n}\in \mathbb {R} _{>0}$, then a closed expression for the distribution of $X=\sum _{i=1}^{n}a_{i}X_{i}$ is not known. It may be, however, approximated efficiently using the property of characteristic functions of chi-squared random variables.

Chi-squared distributions

Noncentral chi-squared distribution

Main article: Noncentral chi-squared distribution

The noncentral chi-squared distribution is obtained from the sum of the squares of independent Gaussian random variables having unit variance and nonzero means.

Generalized chi-squared distribution

Main article: Generalized chi-squared distribution

The generalized chi-squared distribution is obtained from the quadratic form z′Az where z is a zero-mean Gaussian vector having an arbitrary covariance matrix, and A is an arbitrary matrix.

Gamma, exponential, and related distributions

The chi-squared distribution $X\sim \chi _{k}^{2}$ is a special case of the gamma distribution, in that $X\sim \Gamma \left({\frac {k}{2}},{\frac {1}{2}}\right)$ using the rate parameterization of the gamma distribution (or $X\sim \Gamma \left({\frac {k}{2}},2\right)$ using the scale parameterization of the gamma distribution) where k is an integer.

Because the exponential distribution is also a special case of the gamma distribution, we also have that if $$

 
   
     
       X
       ∼
       
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           2
         
         
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   {\displaystyle X\sim \chi _{2}^{2}}
 

, then $$

 
   
     
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       ⁡
       
         (
         
           
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   {\displaystyle X\sim \operatorname {Exp} \left({\frac {1}{2}}\right)}
 

is an exponential distribution.

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       X
       ∼
       
         χ
         
           k
         
         
           2
         
       
     
   
   {\displaystyle X\sim \chi _{k}^{2}}
 

with even k, then X is Erlang distributed with shape parameter k/2 and scale parameter 1/2.

Occurrence and applications

The chi-squared distribution has numerous applications in inferential statistics, for instance in chi-squared tests and in estimating variances. It enters the problem of estimating the mean of a normally distributed population and the problem of estimating the slope of a 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 variables, 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 X₁, ..., X_n are i.i.d. N(μ, σ²) random variables, then $\sum _{i=1}^{n}(X_{i}-{\bar {X}})^{2}\sim \sigma ^{2}\chi _{n-1}^{2}$ where ${\bar {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}$.
• The box below shows some statistics based on X_i ∼ Normal(μ_i, σ²_i), i = 1, ⋯, k, independent random variables that have probability distributions related to the chi-squared distribution:

Name	Statistic
chi-squared distribution	$$ ∑ i = 1 k ( X i − μ i σ i ) 2 {\displaystyle \sum _{i=1}^{k}\left({\frac {X_{i}-\mu _{i}}{\sigma _{i}}}\right)^{2}}
noncentral chi-squared distribution	$$ ∑ i = 1 k ( X i σ i ) 2 {\displaystyle \sum _{i=1}^{k}\left({\frac {X_{i}}{\sigma _{i}}}\right)^{2}}
chi distribution	$$ ∑ i = 1 k ( X i − μ i σ i ) 2 {\displaystyle {\sqrt {\sum _{i=1}^{k}\left({\frac {X_{i}-\mu _{i}}{\sigma _{i}}}\right)^{2}}}}
noncentral chi distribution	$$ ∑ i = 1 k ( X i σ i ) 2 {\displaystyle {\sqrt {\sum _{i=1}^{k}\left({\frac {X_{i}}{\sigma _{i}}}\right)^{2}}}}

The chi-squared distribution is also often encountered in magnetic resonance imaging.

Table of χ² values vs p-values

The p-value is the probability of observing a test statistic at least as extreme in a chi-squared distribution. Accordingly, since the cumulative distribution function (CDF) for the appropriate degrees of freedom (df) gives the probability of having obtained a value less extreme than this point, subtracting the CDF value from 1 gives the p-value. A low p-value, below the chosen significance level, indicates statistical significance, i.e., sufficient evidence to reject the null hypothesis. A significance level of 0.05 is often used as the cutoff between significant and not-significant results.

The table below gives a number of p-values matching to χ² for the first 10 degrees of freedom.

Degrees of freedom (df)	χ2 value
1	0.004	0.02	0.06	0.15	0.46	1.07	1.64	2.71	3.84	6.63	10.83
2	0.10	0.21	0.45	0.71	1.39	2.41	3.22	4.61	5.99	9.21	13.82
3	0.35	0.58	1.01	1.42	2.37	3.66	4.64	6.25	7.81	11.34	16.27
4	0.71	1.06	1.65	2.20	3.36	4.88	5.99	7.78	9.49	13.28	18.47
5	1.14	1.61	2.34	3.00	4.35	6.06	7.29	9.24	11.07	15.09	20.52
6	1.63	2.20	3.07	3.83	5.35	7.23	8.56	10.64	12.59	16.81	22.46
7	2.17	2.83	3.82	4.67	6.35	8.38	9.80	12.02	14.07	18.48	24.32
8	2.73	3.49	4.59	5.53	7.34	9.52	11.03	13.36	15.51	20.09	26.12
9	3.32	4.17	5.38	6.39	8.34	10.66	12.24	14.68	16.92	21.67	27.88
10	3.94	4.87	6.18	7.27	9.34	11.78	13.44	15.99	18.31	23.21	29.59
P value (Probability)	0.95	0.90	0.80	0.70	0.50	0.30	0.20	0.10	0.05	0.01	0.001

These values can be calculated evaluating the quantile function (also known as “inverse CDF” or “ICDF”) of the chi-squared distribution; e. g., the χ² ICDF for p = 0.05 and df = 7 yields 14.06714 ≈ 14.07 as in the table above.

History and name

This distribution was first described by the German statistician Friedrich Robert Helmert in papers of 1875–6, where he computed the sampling distribution of the sample variance of a normal population. Thus in German this was traditionally known as the Helmert'sche ("Helmertian") or "Helmert distribution".

The distribution was independently rediscovered by the English mathematician Karl Pearson in the context of goodness of fit, for which he developed his Pearson's chi-squared test, published in 1900, with computed table of values published in (Elderton 1902), collected in (Pearson 1914, pp. xxxi–xxxiii, 26–28, Table XII).

The name "chi-squared" ultimately derives from Pearson's shorthand for the exponent in a multivariate normal distribution with the Greek letter Chi, writing −½χ² for what would appear in modern notation as −½x^TΣ⁻¹x (Σ being the covariance matrix). The idea of a family of "chi-squared distributions", however, is not due to Pearson but arose as a further development due to Fisher in the 1920s.

See also

• Chi distribution
• Cochran's theorem
• F-distribution
• Fisher's method for combining independent tests of significance
• Gamma distribution
• Generalized chi-squared distribution
• Hotelling's T-squared distribution
• Noncentral chi-squared distribution
• Pearson's chi-squared test
• Reduced chi-squared statistic
• Student's t-distribution
• Wilks's lambda distribution
• Wishart distribution

Further reading

• Hald, Anders (1998). A history of mathematical statistics from 1750 to 1930. New York: Wiley. ISBN 0-471-17912-4.
• Elderton, William Palin (1902). "Tables for Testing the Goodness of Fit of Theory to Observation". Biometrika. 1 (2): 155–163. doi:10.1093/biomet/1.2.155.
• Hazewinkel, Michiel, ed. (2001) [1994], "Chi-squared distribution", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

External links

• Earliest Uses of Some of the Words of Mathematics: entry on Chi squared has a brief history
• Course notes on Chi-Squared Goodness of Fit Testing from Yale University Stats 101 class.
• Mathematica demonstration showing the chi-squared sampling distribution of various statistics, e. g. Σx², for a normal population
• Simple algorithm for approximating cdf and inverse cdf for the chi-squared distribution with a pocket calculator

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   <annotation encoding="application/x-tex">{\displaystyle f(x;\,k)={\begin{cases}{\dfrac {x^{{\frac {k}{2}}-1}e^{-{\frac {x}{2}}}}{2^{\frac {k}{2}}\Gamma \left({\frac {k}{2}}\right)}},&x>0;\<add_contents_exp><m=6 date=20141010>,&{\text{otherwise}}.\end{cases}}}</annotation>
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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Gamma (k/2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \Gamma (k/2)}</annotation> </semantics> </math> denotes the gamma function, which has closed-form values for integer k.

For derivations of the pdf in the cases of one, two and k degrees of freedom, see Proofs related to chi-squared distribution.

Cumulative distribution function

Chernoff bound for the CDF and tail (1-CDF) of a chi-squared random variable with ten degrees of freedom (k = 10)

Its cumulative distribution function is:

<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x;\,k)={\frac {\gamma ({\frac {k}{2}},\,{\frac {x}{2}})}{\Gamma ({\frac {k}{2}})}}=P\left({\frac {k}{2}},\,{\frac {x}{2}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mspace width="thinmathspace" /> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>γ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x;\,k)={\frac {\gamma ({\frac {k}{2}},\,{\frac {x}{2}})}{\Gamma ({\frac {k}{2}})}}=P\left({\frac {k}{2}},\,{\frac {x}{2}}\right),}</annotation> </semantics> </math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (s,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma (s,t)}</annotation> </semantics> </math> is the lower incomplete gamma function and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle P(s,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle P(s,t)}</annotation> </semantics> </math> is the regularized gamma function.

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and the integer recurrence of the gamma function makes it easy to compute for other small even k.

Tables of the chi-squared cumulative distribution function are widely available and the function is included in many spreadsheets and all statistical packages.

Letting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\equiv x/k}">

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</math> (which include all of the cases when this CDF is less than half):

<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(zk;\,k)\leq (ze^{1-z})^{k/2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mi>k</mi> <mo>;</mo> <mspace width="thinmathspace" /> <mi>k</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mo stretchy="false">(</mo> <mi>z</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>z</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(zk;\,k)\leq (ze^{1-z})^{k/2}.}</annotation> </semantics> </math>

The tail bound for the cases when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z>1}</annotation> </semantics> </math>, similarly, is

<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-F(zk;\,k)\leq (ze^{1-z})^{k/2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mi>k</mi> <mo>;</mo> <mspace width="thinmathspace" /> <mi>k</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mo stretchy="false">(</mo> <mi>z</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>z</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-F(zk;\,k)\leq (ze^{1-z})^{k/2}.}</annotation> </semantics> </math>

For another approximation for the CDF modeled after the cube of a Gaussian, see under Noncentral chi-squared distribution.

Additivity

It follows from the definition of the chi-squared distribution that the sum of independent chi-squared variables is also chi-squared distributed. Specifically, if {X_i}_i=1ⁿ are independent chi-squared variables with {k_i}_i=1ⁿ degrees of freedom, respectively, then Y = X₁ + ⋯ + X_n is chi-squared distributed with k₁ + ⋯ + k_n degrees of freedom.

Sample mean

The sample mean of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math> i.i.d. chi-squared variables of degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math> is distributed according to a gamma distribution with shape <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math> and scale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math> parameters:

<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}\sim \operatorname {Gamma} \left(\alpha =n\,k/2,\theta =2/n\right)\qquad {\text{where }}X_{i}\sim \chi ^{2}(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∼</mo> <mi>Gamma</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mo>=</mo> <mi>n</mi> <mspace width="thinmathspace" /> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> <mi>θ</mi> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> <mo>)</mo> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>where </mtext> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∼</mo> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}\sim \operatorname {Gamma} \left(\alpha =n\,k/2,\theta =2/n\right)\qquad {\text{where }}X_{i}\sim \chi ^{2}(k)}</annotation> </semantics> </math>

Asymptotically, given that for a scale parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math> going to infinity, a Gamma distribution converges towards a normal distribution with expectation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =\alpha \cdot \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mi>α</mi> <mo>⋅</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =\alpha \cdot \theta }</annotation> </semantics> </math> and variance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}=\alpha \,\theta ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>α</mi> <mspace width="thinmathspace" /> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}=\alpha \,\theta ^{2}}</annotation> </semantics> </math>, the sample mean converges towards:

<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {X}}{\xrightarrow {n\to \infty }}N(\mu =k,\sigma ^{2}=2\,k/n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mpadded> </mover> </mrow> <mi>N</mi> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>=</mo> <mi>k</mi> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mspace width="thinmathspace" /> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {X}}{\xrightarrow {n\to \infty }}N(\mu =k,\sigma ^{2}=2\,k/n)}</annotation> </semantics> </math>

Note that we would have obtained the same result invoking instead the central limit theorem, noting that for each chi-squared variable of degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math> the expectation is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math> , and its variance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\,k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mspace width="thinmathspace" /> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\,k}</annotation> </semantics> </math> (and hence the variance of the sample mean <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {X}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {X}}}</annotation> </semantics> </math> being <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}=2\,k/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mspace width="thinmathspace" /> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}=2\,k/n}</annotation> </semantics> </math>).

Entropy

The differential entropy is given by

<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=\int _{0}^{\infty }f(x;\,k)\ln f(x;\,k)\,dx={\frac {k}{2}}+\ln \left[2\,\Gamma \left({\frac {k}{2}}\right)\right]+\left(1-{\frac {k}{2}}\right)\,\psi \!\left[{\frac {k}{2}}\right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mspace width="thinmathspace" /> <mi>k</mi> <mo stretchy="false">)</mo> <mi>ln</mi> <mo>⁡</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mspace width="thinmathspace" /> <mi>k</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mn>2</mn> <mspace width="thinmathspace" /> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>ψ</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h=\int _{0}^{\infty }f(x;\,k)\ln f(x;\,k)\,dx={\frac {k}{2}}+\ln \left[2\,\Gamma \left({\frac {k}{2}}\right)\right]+\left(1-{\frac {k}{2}}\right)\,\psi \!\left[{\frac {k}{2}}\right],}</annotation> </semantics> </math>

where ψ(x) is the Digamma function.

The chi-squared distribution is the maximum entropy probability distribution for a random variate X for which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} (X)=k}">

 <semantics>
   <mrow class="MJX-TeXAtom-ORD">
     <mstyle displaystyle="true" scriptlevel="0">
       <mi mathvariant="normal">E</mi>
       <mo>⁡</mo>
       <mo stretchy="false">(</mo>
       <mi>X</mi>
       <mo stretchy="false">)</mo>
       <mo>=</mo>
       <mi>k</mi>
     </mstyle>
   </mrow>
   <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (X)=k}</annotation>
 </semantics>

</math> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} (\ln(X))=\psi (k/2)+\ln(2)}">

 <semantics>
   <mrow class="MJX-TeXAtom-ORD">
     <mstyle displaystyle="true" scriptlevel="0">
       <mi mathvariant="normal">E</mi>
       <mo>⁡</mo>
       <mo stretchy="false">(</mo>
       <mi>ln</mi>
       <mo>⁡</mo>
       <mo stretchy="false">(</mo>
       <mi>X</mi>
       <mo stretchy="false">)</mo>
       <mo stretchy="false">)</mo>
       <mo>=</mo>
       <mi>ψ</mi>
       <mo stretchy="false">(</mo>
       <mi>k</mi>
       <mrow class="MJX-TeXAtom-ORD">
         <mo>/</mo>
       </mrow>
       <mn>2</mn>
       <mo stretchy="false">)</mo>
       <mo>+</mo>
       <mi>ln</mi>
       <mo>⁡</mo>
       <mo stretchy="false">(</mo>
       <mn>2</mn>
       <mo stretchy="false">)</mo>
     </mstyle>
   </mrow>
   <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (\ln(X))=\psi (k/2)+\ln(2)}</annotation>
 </semantics>

</math> are fixed. Since the chi-squared is in the family of gamma distributions, this can be derived by substituting appropriate values in the Expectation of the log moment of gamma. For derivation from more basic principles, see the derivation in moment-generating function of the sufficient statistic.

Noncentral moments

The moments about zero of a chi-squared distribution with k degrees of freedom are given by

<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} (X^{m})=k(k+2)(k+4)\cdots (k+2m-2)=2^{m}{\frac {\Gamma \left(m+{\frac {k}{2}}\right)}{\Gamma \left({\frac {k}{2}}\right)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mi>m</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow> <mi>m</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (X^{m})=k(k+2)(k+4)\cdots (k+2m-2)=2^{m}{\frac {\Gamma \left(m+{\frac {k}{2}}\right)}{\Gamma \left({\frac {k}{2}}\right)}}.}</annotation> </semantics> </math>

Cumulants

The cumulants are readily obtained by a (formal) power series expansion of the logarithm of the characteristic function:

<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa _{n}=2^{n-1}(n-1)!\,k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mspace width="thinmathspace" /> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa _{n}=2^{n-1}(n-1)!\,k}</annotation> </semantics> </math>

Asymptotic properties

By the central limit theorem, because the chi-squared distribution is the sum of k independent random variables with finite mean and variance, it converges to a normal distribution for large k. For many practical purposes, for k > 50 the distribution is sufficiently close to a normal distribution for the difference to be ignored. Specifically, if X ~ χ²(k), then as k tends to infinity, the distribution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X-k)/{\sqrt {2k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>k</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X-k)/{\sqrt {2k}}}</annotation> </semantics> </math> tends to a standard normal distribution. However, convergence is slow as the skewness is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {8/k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {8/k}}}</annotation> </semantics> </math> and the excess kurtosis is 12/k.

The sampling distribution of ln(χ²) converges to normality much faster than the sampling distribution of χ², as the logarithm removes much of the asymmetry. Other functions of the chi-squared distribution converge more rapidly to a normal distribution. Some examples are:

• If X ~ χ²(k) then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\sqrt {2X}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>X</mi> </msqrt> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\sqrt {2X}}}</annotation> </semantics> </math> is approximately normally distributed with mean <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\sqrt {2k-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\sqrt {2k-1}}}</annotation> </semantics> </math> and unit variance (1922, by R. A. Fisher, see (18.23), p. 426 of.
• If X ~ χ²(k) then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\sqrt[{3}]{X/k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\sqrt[{3}]{X/k}}}</annotation> </semantics> </math> is approximately normally distributed with mean <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle 1-2/(9k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>9</mn> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle 1-2/(9k)}</annotation> </semantics> </math> and variance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle 2/(9k).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>9</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle 2/(9k).}</annotation> </semantics> </math> This is known as the Wilson–Hilferty transformation, see (18.24), p. 426 of.

Relation to other distributions

<tbody></tbody>

Approximate formula for median compared with numerical quantile (top). Difference between numerical quantile and approximate formula (bottom).

• As <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\to \infty }</annotation> </semantics> </math>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\chi _{k}^{2}-k)/{\sqrt {2k}}~{\xrightarrow {d}}\ N(0,1)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msubsup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>k</mi> </msqrt> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <mi>d</mi> </mpadded> </mover> </mrow> <mtext> </mtext> <mi>N</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\chi _{k}^{2}-k)/{\sqrt {2k}}~{\xrightarrow {d}}\ N(0,1)\,}</annotation> </semantics> </math> (normal distribution)
• <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{k}^{2}\sim {\chi '}_{k}^{2}(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>∼</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>χ</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{k}^{2}\sim {\chi '}_{k}^{2}(0)}</annotation> </semantics> </math> (noncentral chi-squared distribution with non-centrality parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda =0}</annotation> </semantics> </math>)
• If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\sim \mathrm {F} (\nu _{1},\nu _{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\sim \mathrm {F} (\nu _{1},\nu _{2})}</annotation> </semantics> </math> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\lim _{\nu _{2}\to \infty }\nu _{1}Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\lim _{\nu _{2}\to \infty }\nu _{1}Y}</annotation> </semantics> </math> has the chi-squared distribution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{\nu _{1}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{\nu _{1}}^{2}}</annotation> </semantics> </math>

• As a special case, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\sim \mathrm {F} (1,\nu _{2})\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\sim \mathrm {F} (1,\nu _{2})\,}</annotation> </semantics> </math> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\lim _{\nu _{2}\to \infty }Y\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>Y</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\lim _{\nu _{2}\to \infty }Y\,}</annotation> </semantics> </math> has the chi-squared distribution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{1}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{1}^{2}}</annotation> </semantics> </math>

• <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|{\boldsymbol {N}}_{i=1,\ldots ,k}(0,1)\|^{2}\sim \chi _{k}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>∼</mo> <msubsup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|{\boldsymbol {N}}_{i=1,\ldots ,k}(0,1)\|^{2}\sim \chi _{k}^{2}}</annotation> </semantics> </math> (The squared norm of k standard normally distributed variables is a chi-squared distribution with k degrees of freedom)
• If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \chi ^{2}(\nu )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \chi ^{2}(\nu )\,}</annotation> </semantics> </math> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c>0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>></mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c>0\,}</annotation> </semantics> </math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle cX\sim \Gamma (k=\nu /2,\theta =2c)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mi>X</mi> <mo>∼</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>=</mo> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> <mi>θ</mi> <mo>=</mo> <mn>2</mn> <mi>c</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle cX\sim \Gamma (k=\nu /2,\theta =2c)\,}</annotation> </semantics> </math>. (gamma distribution)
• If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \chi _{k}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <msubsup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \chi _{k}^{2}}</annotation> </semantics> </math> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {X}}\sim \chi _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>X</mi> </msqrt> </mrow> <mo>∼</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {X}}\sim \chi _{k}}</annotation> </semantics> </math> (chi distribution)
• If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \chi ^{2}(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \chi ^{2}(2)}</annotation> </semantics> </math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \operatorname {Exp} (1/2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi>Exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \operatorname {Exp} (1/2)}</annotation> </semantics> </math> is an exponential distribution. (See gamma distribution for more.)
• If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \operatorname {Rayleigh} (1)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi>Rayleigh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \operatorname {Rayleigh} (1)\,}</annotation> </semantics> </math> (Rayleigh distribution) then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{2}\sim \chi ^{2}(2)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>∼</mo> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{2}\sim \chi ^{2}(2)\,}</annotation> </semantics> </math>
• If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \operatorname {Maxwell} (1)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi>Maxwell</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \operatorname {Maxwell} (1)\,}</annotation> </semantics> </math> (Maxwell distribution) then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{2}\sim \chi ^{2}(3)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>∼</mo> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{2}\sim \chi ^{2}(3)\,}</annotation> </semantics> </math>
• If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \chi ^{2}(\nu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \chi ^{2}(\nu )}</annotation> </semantics> </math> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{X}}\sim \operatorname {Inv-} \chi ^{2}(\nu )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>X</mi> </mfrac> </mstyle> </mrow> <mo>∼</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">v</mi> <mtext>-</mtext> </mrow> <mo>⁡</mo> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{X}}\sim \operatorname {Inv-} \chi ^{2}(\nu )\,}</annotation> </semantics> </math> (Inverse-chi-squared distribution)
• The chi-squared distribution is a special case of type 3 Pearson distribution
• If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \chi ^{2}(\nu _{1})\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \chi ^{2}(\nu _{1})\,}</annotation> </semantics> </math> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\sim \chi ^{2}(\nu _{2})\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>∼</mo> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\sim \chi ^{2}(\nu _{2})\,}</annotation> </semantics> </math> are independent then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {X}{X+Y}}\sim \operatorname {Beta} ({\tfrac {\nu _{1}}{2}},{\tfrac {\nu _{2}}{2}})\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>X</mi> <mrow> <mi>X</mi> <mo>+</mo> <mi>Y</mi> </mrow> </mfrac> </mstyle> </mrow> <mo>∼</mo> <mi>Beta</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {X}{X+Y}}\sim \operatorname {Beta} ({\tfrac {\nu _{1}}{2}},{\tfrac {\nu _{2}}{2}})\,}</annotation> </semantics> </math> (beta distribution)
• If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \operatorname {U} (0,1)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi mathvariant="normal">U</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \operatorname {U} (0,1)\,}</annotation> </semantics> </math> (uniform distribution) then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -2\log(X)\sim \chi ^{2}(2)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>2</mn> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -2\log(X)\sim \chi ^{2}(2)\,}</annotation> </semantics> </math>
• <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi ^{2}(6)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>6</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi ^{2}(6)\,}</annotation> </semantics> </math> is a transformation of Laplace distribution
• If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}\sim \operatorname {Laplace} (\mu ,\beta )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∼</mo> <mi>Laplace</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}\sim \operatorname {Laplace} (\mu ,\beta )\,}</annotation> </semantics> </math> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}{\frac {2|X_{i}-\mu |}{\beta }}\sim \chi ^{2}(2n)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mi>β</mi> </mfrac> </mrow> <mo>∼</mo> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}{\frac {2|X_{i}-\mu |}{\beta }}\sim \chi ^{2}(2n)\,}</annotation> </semantics> </math>
• If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation> </semantics> </math> follows the generalized normal distribution (version 1) with parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ,\alpha ,\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>,</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ,\alpha ,\beta }</annotation> </semantics> </math> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}{\frac {2|X_{i}-\mu |^{\beta }}{\alpha }}\sim \chi ^{2}\left({\frac {2n}{\beta }}\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mrow> <mi>α</mi> </mfrac> </mrow> <mo>∼</mo> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mi>β</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}{\frac {2|X_{i}-\mu |^{\beta }}{\alpha }}\sim \chi ^{2}\left({\frac {2n}{\beta }}\right)\,}</annotation> </semantics> </math>
• chi-squared distribution is a transformation of Pareto distribution
• Student's t-distribution is a transformation of chi-squared distribution
• Student's t-distribution can be obtained from chi-squared distribution and normal distribution
• Noncentral beta distribution can be obtained as a transformation of chi-squared distribution and Noncentral chi-squared distribution
• Noncentral t-distribution can be obtained from normal distribution and chi-squared distribution

A chi-squared variable with k degrees of freedom is defined as the sum of the squares of k independent standard normal random variables.

If Y is a k-dimensional Gaussian random vector with mean vector μ and rank k covariance matrix C, then X = (Y−μ)^TC⁻¹(Y − μ) is chi-squared distributed with k degrees of freedom.

The sum of squares of statistically independent unit-variance Gaussian variables which do not have mean zero yields a generalization of the chi-squared distribution called the noncentral chi-squared distribution.

If Y is a vector of k i.i.d. standard normal random variables and A is a k×k symmetric, idempotent matrix with rank k−n then the quadratic form Y^TAY is chi-squared distributed with k−n degrees of freedom.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

 <semantics>
   <mrow class="MJX-TeXAtom-ORD">
     <mstyle displaystyle="true" scriptlevel="0">
       <mi mathvariant="normal">Σ</mi>
     </mstyle>
   </mrow>
   <annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>
 </semantics>

</math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\times p}">

 <semantics>
   <mrow class="MJX-TeXAtom-ORD">
     <mstyle displaystyle="true" scriptlevel="0">
       <mi>p</mi>
       <mo>×</mo>
       <mi>p</mi>
     </mstyle>
   </mrow>
   <annotation encoding="application/x-tex">{\displaystyle p\times p}</annotation>
 </semantics>

</math> positive-semidefinite covariance matrix with strictly positive diagonal entries, then for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim N(0,\Sigma )}">

 <semantics>
   <mrow class="MJX-TeXAtom-ORD">
     <mstyle displaystyle="true" scriptlevel="0">
       <mi>X</mi>
       <mo>∼</mo>
       <mi>N</mi>
       <mo stretchy="false">(</mo>
       <mn>0</mn>
       <mo>,</mo>
       <mi mathvariant="normal">Σ</mi>
       <mo stretchy="false">)</mo>
     </mstyle>
   </mrow>
   <annotation encoding="application/x-tex">{\displaystyle X\sim N(0,\Sigma )}</annotation>
 </semantics>

</math> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}">

 <semantics>
   <mrow class="MJX-TeXAtom-ORD">
     <mstyle displaystyle="true" scriptlevel="0">
       <mi>w</mi>
     </mstyle>
   </mrow>
   <annotation encoding="application/x-tex">{\displaystyle w}</annotation>
 </semantics>

</math> a random <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

 <semantics>
   <mrow class="MJX-TeXAtom-ORD">
     <mstyle displaystyle="true" scriptlevel="0">
       <mi>p</mi>
     </mstyle>
   </mrow>
   <annotation encoding="application/x-tex">{\displaystyle p}</annotation>
 </semantics>

</math>-vector independent of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

 <semantics>
   <mrow class="MJX-TeXAtom-ORD">
     <mstyle displaystyle="true" scriptlevel="0">
       <mi>X</mi>
     </mstyle>
   </mrow>
   <annotation encoding="application/x-tex">{\displaystyle X}</annotation>
 </semantics>

</math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{1}+\cdots +w_{p}=1}">

 <semantics>
   <mrow class="MJX-TeXAtom-ORD">
     <mstyle displaystyle="true" scriptlevel="0">
       <msub>
         <mi>w</mi>
         <mrow class="MJX-TeXAtom-ORD">
           <mn>1</mn>
         </mrow>
       </msub>
       <mo>+</mo>
       <mo>⋯</mo>
       <mo>+</mo>
       <msub>
         <mi>w</mi>
         <mrow class="MJX-TeXAtom-ORD">
           <mi>p</mi>
         </mrow>
       </msub>
       <mo>=</mo>
       <mn>1</mn>
     </mstyle>
   </mrow>
   <annotation encoding="application/x-tex">{\displaystyle w_{1}+\cdots +w_{p}=1}</annotation>
 </semantics>

</math> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{i}\geq 0,i=1,\cdots ,p,}">

 <semantics>
   <mrow class="MJX-TeXAtom-ORD">
     <mstyle displaystyle="true" scriptlevel="0">
       <msub>
         <mi>w</mi>
         <mrow class="MJX-TeXAtom-ORD">
           <mi>i</mi>
         </mrow>
       </msub>
       <mo>≥</mo>
       <mn>0</mn>
       <mo>,</mo>
       <mi>i</mi>
       <mo>=</mo>
       <mn>1</mn>
       <mo>,</mo>
       <mo>⋯</mo>
       <mo>,</mo>
       <mi>p</mi>
       <mo>,</mo>
     </mstyle>
   </mrow>
   <annotation encoding="application/x-tex">{\displaystyle w_{i}\geq 0,i=1,\cdots ,p,}</annotation>
 </semantics>

</math> it holds that

<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\left({\frac {w_{1}}{X_{1}}},\cdots ,{\frac {w_{p}}{X_{p}}}\right)\Sigma \left({\frac {w_{1}}{X_{1}}},\cdots ,{\frac {w_{p}}{X_{p}}}\right)^{\top }}}\sim \chi _{1}^{2}.}">

 <semantics>
   <mrow class="MJX-TeXAtom-ORD">
     <mstyle displaystyle="true" scriptlevel="0">
       <mrow class="MJX-TeXAtom-ORD">
         <mfrac>
           <mn>1</mn>
           <mrow>
             <mrow>
               <mo>(</mo>
               <mrow>
                 <mrow class="MJX-TeXAtom-ORD">
                   <mfrac>
                     <msub>
                       <mi>w</mi>
                       <mrow class="MJX-TeXAtom-ORD">
                         <mn>1</mn>
                       </mrow>
                     </msub>
                     <msub>
                       <mi>X</mi>
                       <mrow class="MJX-TeXAtom-ORD">
                         <mn>1</mn>
                       </mrow>
                     </msub>
                   </mfrac>
                 </mrow>
                 <mo>,</mo>
                 <mo>⋯</mo>
                 <mo>,</mo>
                 <mrow class="MJX-TeXAtom-ORD">
                   <mfrac>
                     <msub>
                       <mi>w</mi>
                       <mrow class="MJX-TeXAtom-ORD">
                         <mi>p</mi>
                       </mrow>
                     </msub>
                     <msub>
                       <mi>X</mi>
                       <mrow class="MJX-TeXAtom-ORD">
                         <mi>p</mi>
                       </mrow>
                     </msub>
                   </mfrac>
                 </mrow>
               </mrow>
               <mo>)</mo>
             </mrow>
             <mi mathvariant="normal">Σ</mi>
             <msup>
               <mrow>
                 <mo>(</mo>
                 <mrow>
                   <mrow class="MJX-TeXAtom-ORD">
                     <mfrac>
                       <msub>
                         <mi>w</mi>
                         <mrow class="MJX-TeXAtom-ORD">
                           <mn>1</mn>
                         </mrow>
                       </msub>
                       <msub>
                         <mi>X</mi>
                         <mrow class="MJX-TeXAtom-ORD">
                           <mn>1</mn>
                         </mrow>
                       </msub>
                     </mfrac>
                   </mrow>
                   <mo>,</mo>
                   <mo>⋯</mo>
                   <mo>,</mo>
                   <mrow class="MJX-TeXAtom-ORD">
                     <mfrac>
                       <msub>
                         <mi>w</mi>
                         <mrow class="MJX-TeXAtom-ORD">
                           <mi>p</mi>
                         </mrow>
                       </msub>
                       <msub>
                         <mi>X</mi>
                         <mrow class="MJX-TeXAtom-ORD">
                           <mi>p</mi>
                         </mrow>
                       </msub>
                     </mfrac>
                   </mrow>
                 </mrow>
                 <mo>)</mo>
               </mrow>
               <mrow class="MJX-TeXAtom-ORD">
                 <mi mathvariant="normal">⊤</mi>
               </mrow>
             </msup>
           </mrow>
         </mfrac>
       </mrow>
       <mo>∼</mo>
       <msubsup>
         <mi>χ</mi>
         <mrow class="MJX-TeXAtom-ORD">
           <mn>1</mn>
         </mrow>
         <mrow class="MJX-TeXAtom-ORD">
           <mn>2</mn>
         </mrow>
       </msubsup>
       <mo>.</mo>
     </mstyle>
   </mrow>
   <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\left({\frac {w_{1}}{X_{1}}},\cdots ,{\frac {w_{p}}{X_{p}}}\right)\Sigma \left({\frac {w_{1}}{X_{1}}},\cdots ,{\frac {w_{p}}{X_{p}}}\right)^{\top }}}\sim \chi _{1}^{2}.}</annotation>
 </semantics>

</math>

The chi-squared distribution is also naturally related to other distributions arising from the Gaussian. In particular,

• Y is F-distributed, Y ~ F(k₁,k₂) if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle Y={\frac {X_{1}/k_{1}}{X_{2}/k_{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>Y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle Y={\frac {X_{1}/k_{1}}{X_{2}/k_{2}}}}</annotation> </semantics> </math> where X₁ ~ χ²(k₁) and X₂  ~ χ²(k₂) are statistically independent.
• If X₁  ~  χ²_k1 and X₂  ~  χ²_k2 are statistically independent, then X₁ + X₂  ~ χ²_k1+k2. If X₁ and X₂ are not independent, then X₁ + X₂ is not chi-squared distributed.

Generalizations

The chi-squared distribution is obtained as the sum of the squares of k independent, zero-mean, unit-variance Gaussian random variables. Generalizations of this distribution can be obtained by summing the squares of other types of Gaussian random variables. Several such distributions are described below.

Linear combination

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},\ldots ,X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},\ldots ,X_{n}}</annotation> </semantics> </math> are chi square random variables and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},\ldots ,a_{n}\in \mathbb {R} _{>0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>></mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},\ldots ,a_{n}\in \mathbb {R} _{>0}}</annotation> </semantics> </math>, then a closed expression for the distribution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\sum _{i=1}^{n}a_{i}X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\sum _{i=1}^{n}a_{i}X_{i}}</annotation> </semantics> </math> is not known. It may be, however, approximated efficiently using the property of characteristic functions of chi-squared random variables.

Chi-squared distributions

Noncentral chi-squared distribution

Main article: Noncentral chi-squared distribution

The noncentral chi-squared distribution is obtained from the sum of the squares of independent Gaussian random variables having unit variance and nonzero means.

Generalized chi-squared distribution

Main article: Generalized chi-squared distribution

The generalized chi-squared distribution is obtained from the quadratic form z′Az where z is a zero-mean Gaussian vector having an arbitrary covariance matrix, and A is an arbitrary matrix.

Gamma, exponential, and related distributions

The chi-squared distribution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \chi _{k}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <msubsup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \chi _{k}^{2}}</annotation> </semantics> </math> is a special case of the gamma distribution, in that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \Gamma \left({\frac {k}{2}},{\frac {1}{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \Gamma \left({\frac {k}{2}},{\frac {1}{2}}\right)}</annotation> </semantics> </math> using the rate parameterization of the gamma distribution (or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \Gamma \left({\frac {k}{2}},2\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \Gamma \left({\frac {k}{2}},2\right)}</annotation> </semantics> </math> using the scale parameterization of the gamma distribution) where k is an integer.

Because the exponential distribution is also a special case of the gamma distribution, we also have that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \chi _{2}^{2}}">

 <semantics>
   <mrow class="MJX-TeXAtom-ORD">
     <mstyle displaystyle="true" scriptlevel="0">
       <mi>X</mi>
       <mo>∼</mo>
       <msubsup>
         <mi>χ</mi>
         <mrow class="MJX-TeXAtom-ORD">
           <mn>2</mn>
         </mrow>
         <mrow class="MJX-TeXAtom-ORD">
           <mn>2</mn>
         </mrow>
       </msubsup>
     </mstyle>
   </mrow>
   <annotation encoding="application/x-tex">{\displaystyle X\sim \chi _{2}^{2}}</annotation>
 </semantics>

</math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \operatorname {Exp} \left({\frac {1}{2}}\right)}">

 <semantics>
   <mrow class="MJX-TeXAtom-ORD">
     <mstyle displaystyle="true" scriptlevel="0">
       <mi>X</mi>
       <mo>∼</mo>
       <mi>Exp</mi>
       <mo>⁡</mo>
       <mrow>
         <mo>(</mo>
         <mrow class="MJX-TeXAtom-ORD">
           <mfrac>
             <mn>1</mn>
             <mn>2</mn>
           </mfrac>
         </mrow>
         <mo>)</mo>
       </mrow>
     </mstyle>
   </mrow>
   <annotation encoding="application/x-tex">{\displaystyle X\sim \operatorname {Exp} \left({\frac {1}{2}}\right)}</annotation>
 </semantics>

</math> is an exponential distribution.

The Erlang distribution is also a special case of the gamma distribution and thus we also have that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \chi _{k}^{2}}">

 <semantics>
   <mrow class="MJX-TeXAtom-ORD">
     <mstyle displaystyle="true" scriptlevel="0">
       <mi>X</mi>
       <mo>∼</mo>
       <msubsup>
         <mi>χ</mi>
         <mrow class="MJX-TeXAtom-ORD">
           <mi>k</mi>
         </mrow>
         <mrow class="MJX-TeXAtom-ORD">
           <mn>2</mn>
         </mrow>
       </msubsup>
     </mstyle>
   </mrow>
   <annotation encoding="application/x-tex">{\displaystyle X\sim \chi _{k}^{2}}</annotation>
 </semantics>

</math> with even k, then X is Erlang distributed with shape parameter k/2 and scale parameter 1/2.

Occurrence and applications

The chi-squared distribution has numerous applications in inferential statistics, for instance in chi-squared tests and in estimating variances. It enters the problem of estimating the mean of a normally distributed population and the problem of estimating the slope of a 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 variables, 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 X₁, ..., X_n are i.i.d. N(μ, σ²) random variables, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}(X_{i}-{\bar {X}})^{2}\sim \sigma ^{2}\chi _{n-1}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>∼</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}(X_{i}-{\bar {X}})^{2}\sim \sigma ^{2}\chi _{n-1}^{2}}</annotation> </semantics> </math> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}}</annotation> </semantics> </math>.
• The box below shows some statistics based on X_i ∼ Normal(μ_i, σ²_i), i = 1, ⋯, k, independent random variables that have probability distributions related to the chi-squared distribution:

<tbody></tbody>

Name	Statistic
chi-squared distribution	<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{k}\left({\frac {X_{i}-\mu _{i}}{\sigma _{i}}}\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{k}\left({\frac {X_{i}-\mu _{i}}{\sigma _{i}}}\right)^{2}}</annotation> </semantics> </math>
noncentral chi-squared distribution	<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{k}\left({\frac {X_{i}}{\sigma _{i}}}\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{k}\left({\frac {X_{i}}{\sigma _{i}}}\right)^{2}}</annotation> </semantics> </math>
chi distribution	<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\sum _{i=1}^{k}\left({\frac {X_{i}-\mu _{i}}{\sigma _{i}}}\right)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\sum _{i=1}^{k}\left({\frac {X_{i}-\mu _{i}}{\sigma _{i}}}\right)^{2}}}}</annotation> </semantics> </math>
noncentral chi distribution	<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\sum _{i=1}^{k}\left({\frac {X_{i}}{\sigma _{i}}}\right)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\sum _{i=1}^{k}\left({\frac {X_{i}}{\sigma _{i}}}\right)^{2}}}}</annotation> </semantics> </math>

The chi-squared distribution is also often encountered in magnetic resonance imaging.

Table of χ² values vs p-values

The p-value is the probability of observing a test statistic at least as extreme in a chi-squared distribution. Accordingly, since the cumulative distribution function (CDF) for the appropriate degrees of freedom (df) gives the probability of having obtained a value less extreme than this point, subtracting the CDF value from 1 gives the p-value. A low p-value, below the chosen significance level, indicates statistical significance, i.e., sufficient evidence to reject the null hypothesis. A significance level of 0.05 is often used as the cutoff between significant and not-significant results.

The table below gives a number of p-values matching to χ² for the first 10 degrees of freedom.

<tbody> </tbody>

Degrees of freedom (df)	χ2 value
1	0.004	0.02	0.06	0.15	0.46	1.07	1.64	2.71	3.84	6.63	10.83
2	0.10	0.21	0.45	0.71	1.39	2.41	3.22	4.61	5.99	9.21	13.82
3	0.35	0.58	1.01	1.42	2.37	3.66	4.64	6.25	7.81	11.34	16.27
4	0.71	1.06	1.65	2.20	3.36	4.88	5.99	7.78	9.49	13.28	18.47
5	1.14	1.61	2.34	3.00	4.35	6.06	7.29	9.24	11.07	15.09	20.52
6	1.63	2.20	3.07	3.83	5.35	7.23	8.56	10.64	12.59	16.81	22.46
7	2.17	2.83	3.82	4.67	6.35	8.38	9.80	12.02	14.07	18.48	24.32
8	2.73	3.49	4.59	5.53	7.34	9.52	11.03	13.36	15.51	20.09	26.12
9	3.32	4.17	5.38	6.39	8.34	10.66	12.24	14.68	16.92	21.67	27.88
10	3.94	4.87	6.18	7.27	9.34	11.78	13.44	15.99	18.31	23.21	29.59
P value (Probability)	0.95	0.90	0.80	0.70	0.50	0.30	0.20	0.10	0.05	0.01	0.001

These values can be calculated evaluating the quantile function (also known as “inverse CDF” or “ICDF”) of the chi-squared distribution; e. g., the χ² ICDF for p = 0.05 and df = 7 yields 14.06714 ≈ 14.07 as in the table above.

History and name

This distribution was first described by the German statistician Friedrich Robert Helmert in papers of 1875–6, where he computed the sampling distribution of the sample variance of a normal population. Thus in German this was traditionally known as the Helmert'sche ("Helmertian") or "Helmert distribution".

The distribution was independently rediscovered by the English mathematician Karl Pearson in the context of goodness of fit, for which he developed his Pearson's chi-squared test, published in 1900, with computed table of values published in (Elderton 1902), collected in (Pearson 1914, pp. xxxi–xxxiii, 26–28, Table XII).

The name "chi-squared" ultimately derives from Pearson's shorthand for the exponent in a multivariate normal distribution with the Greek letter Chi, writing −½χ² for what would appear in modern notation as −½x^TΣ⁻¹x (Σ being the covariance matrix). The idea of a family of "chi-squared distributions", however, is not due to Pearson but arose as a further development due to Fisher in the 1920s.

See also

• Chi distribution
• Cochran's theorem
• F-distribution
• Fisher's method for combining independent tests of significance
• Gamma distribution
• Generalized chi-squared distribution
• Hotelling's T-squared distribution
• Noncentral chi-squared distribution
• Pearson's chi-squared test
• Reduced chi-squared statistic
• Student's t-distribution
• Wilks's lambda distribution
• Wishart distribution

Further reading

<style data-mw-deduplicate="TemplateStyles:r853264625">.mw-parser-output .refbegin{font-size:90%;margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{list-style-type:none;margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li,.mw-parser-output .refbegin-hanging-indents>dl>dd{margin-left:0;padding-left:3.2em;text-indent:-3.2em;list-style:none}.mw-parser-output .refbegin-100{font-size:100%}</style>

• Hald, Anders (1998). A history of mathematical statistics from 1750 to 1930. New York: Wiley. ISBN 0-471-17912-4.<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r861714446"/>
• Elderton, William Palin (1902). "Tables for Testing the Goodness of Fit of Theory to Observation". Biometrika. 1 (2): 155–163. doi:10.1093/biomet/1.2.155.<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r861714446"/>
• Hazewinkel, Michiel, ed. (2001) [1994], "Chi-squared distribution", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r861714446"/>

External links

• Earliest Uses of Some of the Words of Mathematics: entry on Chi squared has a brief history
• Course notes on Chi-Squared Goodness of Fit Testing from Yale University Stats 101 class.
• Mathematica demonstration showing the chi-squared sampling distribution of various statistics, e. g. Σx², for a normal population
• Simple algorithm for approximating cdf and inverse cdf for the chi-squared distribution with a pocket calculator

</raw>

</toggledisplay>

English Journal

• Functional polymorphisms in the IL6 gene promoter and the risk of urinary bladder cancer in India.

• Gautam KA1, Muktanand T2, Sankhwar SN1, Goel A1, Sankhwar PL3, Rajender S4.
• Cytokine.Cytokine.2016 Jan;77:152-6. doi: 10.1016/j.cyto.2015.11.008. Epub 2015 Nov 19.
• BACKGROUND: Interleukin-6 is a multifunctional cytokine, which plays a key role in tumor proliferation and differentiation. Variations in its gene (IL6) sequence may affect the risk of developing various cancers, including urinary bladder cancer. The present study was done to find the association of
• PMID 26605964

• Eating disorders in a multi-ethnic inner-city UK sample: prevalence, comorbidity and service use.

• Solmi F1,2, Hotopf M3, Hatch SL3, Treasure J4, Micali N5,6.
• Social psychiatry and psychiatric epidemiology.Soc Psychiatry Psychiatr Epidemiol.2015 Dec 2. [Epub ahead of print]
• PURPOSE: No studies have investigated the prevalence of eating disorders (ED) according to DSM-5 criteria and few have explored their comorbidity and service use in the general population in the UK. We aimed to estimate the prevalence, comorbidity, and service use in individuals with ED in a multi-e
• PMID 26631229

• Association between I/D polymorphism in the ACE gene and sarcoidosis in Turkish patients.

• Sarı G1, Kurt E2, Saydam F3, Değirmenci İ4, Güneş HV4.
• Cytotechnology.Cytotechnology.2015 Dec;67(6):1067-72. doi: 10.1007/s10616-014-9747-7. Epub 2014 Jul 10.
• Sarcoidosis is a chronic inflammatory disease with a complex pathogenesis and unknown etiology characterized by noncaseating granulomas that invade the lungs, eyes, liver and other organs. Insertion (I)/deletion (D) polymorphism in the gene encoding the angiotensin-converting enzyme (ACE) has been s
• PMID 25008296

Japanese Journal

• テンプレートマッチングにおける最適なテンプレート形状の生成方法 : 多様性を考慮した遺伝的アルゴリズムによる検討

• 二瓶 俊彦,イミティ アデルジャン,萩原 由香里 [他]
• 画像電子学会誌 The journal of the Institute of Image Electronics Engineers of Japan : visual computing, devices & communications 44(1), 28-38, 2015
• NAID 40020392641

• 蒸気タービン損傷の2変数統計解析による保全時期設定法の検討

• 藤山 一成,飯田 雄大,齊藤 和宏
• 材料 64(3), 223-228, 2015
• … Bivariate log-normal distribution analyses were conducted on the specific output class of steam turbine components such as rotors, moving blades, nozzles, casings. …
• NAID 130004749949

• 固有値分布のχ^2近似と母固有値の信頼区間の構成

• 加藤 仁志,橋口 博樹
• 計算機統計学 27(1), 11-28, 2014-11-01
• 本論文では,固有値分布のχ^2近似に着目し,自由度の異なるχ^2乗分布の積により新たな固有値分布の近似式を提案する.このχ^2乗の積による近似は,これまで知られている単体でのχ^2よりも数値精度の高いことをシミュレーションによる数値実験により示す.さらに,単体でのχ^2とχ^2の積による近似において,両者による母集団固有値の信頼区間の構成も議論する.後者は,全ての母集団固有値を推定した上で信頼区間 …
• NAID 110009865895

Related Links

• Chi Square Distribution - 九州大学数理生物学研究室

[a]自由度1のカイ2乗分布 が標準正規分布 に従う確率変数とする。このとき の従う分布を自由度1のカイ2乗分布と言う。 の分布密度関数 が

• Chi-Square Distribution - Statistics and Probability

All about chi-square probability distribution. How to compute chi-square statistic and chi-square probability. Includes chi-square examples with solutions. ... Chi-Square Distribution Calculator The Chi-Square Distribution ...

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★リンクテーブル★

リンク元	「カイ二乗分布」「χ二乗分布」
関連記事	「distribution」「square」「c」

「カイ二乗分布」

　　[★]

英

chi-square distribution

関

カイ二乗検定、χ二乗分布

「χ二乗分布」

　　[★]

英

chi-square distribution

関

カイ二乗分布

「distribution」

　　[★]

• n.

• 分配、配布、配給、配分。流通、販売。(郵便の)区分。(統計)分布。(薬)分布(証券)売り抜け。(ミサの)聖品の分餐

　 　 　 　 　

「square」

　　[★]

• n.

• 平方、二乗、四角

• adj.

• 四角の

関

sq

　 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　

「c」

　　[★]

• 光速, speed of light