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Source: Chi-squared distribution

In probability theory and statistics, the chi-squared distribution (also chi-square or

χ

2

{\displaystyle \chi ^{2}}

-distribution) with

k

{\displaystyle k}

degrees of freedom is the distribution of a sum of the squares of

k

{\displaystyle k}

independent standard normal random variables.
The chi-squared distribution

χ

k

2

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

is a special case of the gamma distribution and the univariate Wishart distribution. Specifically if

X
∼

χ

k

2

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

then

X
∼

Gamma

(
α
=

k
2

,
θ
=
2
)

{\displaystyle X\sim {\text{Gamma}}(\alpha ={\frac {k}{2}},\theta =2)}

(where

α

{\displaystyle \alpha }

is the shape parameter and

θ

{\displaystyle \theta }

the scale parameter of the gamma distribution) and

X
∼

W

1

(
1
,
k
)

{\displaystyle X\sim {\text{W}}_{1}(1,k)}

.
The scaled chi-squared distribution

s

2

χ

k

2

{\displaystyle s^{2}\chi _{k}^{2}}

is a reparametrization of the gamma distribution and the univariate Wishart distribution. Specifically if

X
∼

s

2

χ

k

2

{\displaystyle X\sim s^{2}\chi _{k}^{2}}

then

X
∼

Gamma

(
α
=

k
2

,
θ
=
2

s

2

)

{\displaystyle X\sim {\text{Gamma}}(\alpha ={\frac {k}{2}},\theta =2s^{2})}

and

X
∼

W

1

(

s

2

,
k
)

{\displaystyle X\sim {\text{W}}_{1}(s^{2},k)}

.
The chi-squared distribution is one of the most widely used probability distributions in inferential statistics, notably in hypothesis testing and in construction of confidence intervals. This distribution is sometimes called the central chi-squared distribution, a special case of the more general noncentral 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 finding the confidence interval for estimating the 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.

Definitions

chi

= Introduction

= Probability density function

= Cumulative distribution function

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

F
(
z
k
;

k
)
≤
(
z

e

1
−
z

)

k

/

2

.

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

The tail bound for the cases when

z
>
1

{\displaystyle z>1}

, similarly, is

1
−
F
(
z
k
;

k
)
≤
(
z

e

1
−
z

)

k

/

2

.

{\displaystyle 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.

Properties

= Cochran's theorem

=

The following is a special case of Cochran's theorem.
Theorem. If

Z

1

,
.
.
.
,

Z

n

{\displaystyle Z_{1},...,Z_{n}}

are independent identically distributed (i.i.d.), standard normal random variables, then

∑

t
=
1

n

(

Z

t

−

Z
¯

)

2

∼

χ

n
−
1

2

{\displaystyle \sum _{t=1}^{n}(Z_{t}-{\bar {Z}})^{2}\sim \chi _{n-1}^{2}}

where

Z
¯

=

1
n

∑

t
=
1

n

Z

t

.

{\displaystyle {\bar {Z}}={\frac {1}{n}}\sum _{t=1}^{n}Z_{t}.}

= 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
=

1
,
n

¯

{\displaystyle X_{i},i={\overline {1,n}}}

are independent chi-squared variables with

k

i

{\displaystyle k_{i}}

,

i
=

1
,
n

¯

{\displaystyle i={\overline {1,n}}}

degrees of freedom, respectively, then

Y
=

X

1

+
⋯
+

X

n

{\displaystyle Y=X_{1}+\cdots +X_{n}}

is chi-squared distributed with

k

1

+
⋯
+

k

n

{\displaystyle k_{1}+\cdots +k_{n}}

degrees of freedom.

= Sample mean

=
The sample mean of

n

{\displaystyle n}

i.i.d. chi-squared variables of degree

k

{\displaystyle k}

is distributed according to a gamma distribution with shape

α

{\displaystyle \alpha }

and scale

θ

{\displaystyle \theta }

parameters:

X
¯

=

1
n

∑

i
=
1

n

X

i

∼
Gamma
⁡

(

α
=
n

k

/

2
,
θ
=
2

/

n

)

where

X

i

∼

χ

2

(
k
)

{\displaystyle {\overline {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 shape parameter

α

{\displaystyle \alpha }

going to infinity, a Gamma distribution converges towards a normal distribution with expectation

μ
=
α
⋅
θ

{\displaystyle \mu =\alpha \cdot \theta }

and variance

σ

2

=
α

θ

2

{\displaystyle \sigma ^{2}=\alpha \,\theta ^{2}}

, the sample mean converges towards:

X
¯

→

n
→
∞

N
(
μ
=
k
,

σ

2

=
2

k

/

n
)

{\displaystyle {\overline {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

{\displaystyle k}

the expectation is

k

{\displaystyle k}

, and its variance

2

k

{\displaystyle 2\,k}

(and hence the variance of the sample mean

X
¯

{\displaystyle {\overline {X}}}

being

σ

2

=

2
k

n

{\displaystyle \sigma ^{2}={\frac {2k}{n}}}

).

= Entropy

=
The differential entropy is given by

h
=

∫

0

∞

f
(
x
;

k
)
ln
⁡
f
(
x
;

k
)

d
x
=

k
2

+
ln
⁡

[

2

Γ

(

k
2

)

]

+

(

1
−

k
2

)

ψ

(

k
2

)

,

{\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),}

where

ψ
(
x
)

{\displaystyle \psi (x)}

is the Digamma function.
The chi-squared distribution is the maximum entropy probability distribution for a random variate

X

{\displaystyle 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 noncentral moments (raw moments) of a chi-squared distribution with

k

{\displaystyle k}

degrees of freedom are given by

E
⁡
(

X

m

)
=
k
(
k
+
2
)
(
k
+
4
)
⋯
(
k
+
2
m
−
2
)
=

2

m

Γ

(

m
+

k
2

)

Γ

(

k
2

)

.

{\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)}}.}

= Cumulants

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

κ

n

=

2

n
−
1

(
n
−
1
)
!

k

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

= Concentration

=
The chi-squared distribution exhibits strong concentration around its mean. The standard Laurent-Massart bounds are:

P
⁡
(
X
−
k
≥
2

k
x

+
2
x
)
≤
exp
⁡
(
−
x
)

{\displaystyle \operatorname {P} (X-k\geq 2{\sqrt {kx}}+2x)\leq \exp(-x)}

P
⁡
(
k
−
X
≥
2

k
x

)
≤
exp
⁡
(
−
x
)

{\displaystyle \operatorname {P} (k-X\geq 2{\sqrt {kx}})\leq \exp(-x)}

One consequence is that, if

v
∼
N
(
0
,
1

)

n

{\displaystyle v\sim N(0,1)^{n}}

is a gaussian random vector in

R

n

{\displaystyle \mathbb {R} ^{n}}

, then as the dimension

n

{\displaystyle n}

grows, the squared length of the vector is concentrated tightly around

n

{\displaystyle n}

with a width

n

1

/

2
+
α

{\displaystyle n^{1/2+\alpha }}

:

P
r
(
‖
v

‖

2

∈
[
n
−
2

n

1

/

2
+
α

,
n
+
2

n

1

/

2
+
α

+
2

n

α

]
)
≥
1
−

e

−

n

α

{\displaystyle Pr(\|v\|^{2}\in [n-2n^{1/2+\alpha },n+2n^{1/2+\alpha }+2n^{\alpha }])\geq 1-e^{-n^{\alpha }}}

where the exponent

α

{\displaystyle \alpha }

can be chosen as any value in

(
0
,
1

/

2
)

{\displaystyle (0,1/2)}

.

= Asymptotic properties

=

By the central limit theorem, because the chi-squared distribution is the sum of

k

{\displaystyle k}

independent random variables with finite mean and variance, it converges to a normal distribution for large

k

{\displaystyle k}

. For many practical purposes, for

k
>
50

{\displaystyle k>50}

the distribution is sufficiently close to a normal distribution, so the difference is ignorable. Specifically, if

X
∼

χ

2

(
k
)

{\displaystyle X\sim \chi ^{2}(k)}

, then as

k

{\displaystyle k}

tends to infinity, the distribution of

(
X
−
k
)

/

2
k

{\displaystyle (X-k)/{\sqrt {2k}}}

tends to a standard normal distribution. However, convergence is slow as the skewness is

8

/

k

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

and the excess kurtosis is

12

/

k

{\displaystyle 12/k}

.
The sampling distribution of

ln
⁡
(

χ

2

)

{\displaystyle \ln(\chi ^{2})}

converges to normality much faster than the sampling distribution of

χ

2

{\displaystyle \chi ^{2}}

, as the logarithmic transform removes much of the asymmetry.
Other functions of the chi-squared distribution converge more rapidly to a normal distribution. Some examples are:

If

X
∼

χ

2

(
k
)

{\displaystyle X\sim \chi ^{2}(k)}

then

2
X

{\displaystyle {\sqrt {2X}}}

is approximately normally distributed with mean

2
k
−
1

{\displaystyle {\sqrt {2k-1}}}

and unit variance (1922, by R. A. Fisher, see (18.23), p. 426 of Johnson.
If

X
∼

χ

2

(
k
)

{\displaystyle X\sim \chi ^{2}(k)}

then

X

/

k

3

{\displaystyle {\sqrt[{3}]{X/k}}}

is approximately normally distributed with mean

1
−

2

9
k

{\displaystyle 1-{\frac {2}{9k}}}

and variance

2

9
k

.

{\displaystyle {\frac {2}{9k}}.}

This is known as the Wilson–Hilferty transformation, see (18.24), p. 426 of Johnson.
This normalizing transformation leads directly to the commonly used median approximation

k

(

1
−

2

9
k

)

3

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

by back-transforming from the mean, which is also the median, of the normal distribution.

Related distributions

As

k
→
∞

{\displaystyle k\to \infty }

,

(

χ

k

2

−
k
)

/

2
k

→

d

N
(
0
,
1
)

{\displaystyle (\chi _{k}^{2}-k)/{\sqrt {2k}}~{\xrightarrow {d}}\ N(0,1)\,}

(normal distribution)

χ

k

2

∼

χ
′

k

2

(
0
)

{\displaystyle \chi _{k}^{2}\sim {\chi '}_{k}^{2}(0)}

(noncentral chi-squared distribution with non-centrality parameter

λ
=
0

{\displaystyle \lambda =0}

)
If

Y
∼

F

(

ν

1

,

ν

2

)

{\displaystyle Y\sim \mathrm {F} (\nu _{1},\nu _{2})}

then

X
=

lim

ν

2

→
∞

ν

1

Y

{\displaystyle X=\lim _{\nu _{2}\to \infty }\nu _{1}Y}

has the chi-squared distribution

χ

ν

1

2

{\displaystyle \chi _{\nu _{1}}^{2}}

As a special case, if

Y
∼

F

(
1
,

ν

2

)

{\displaystyle Y\sim \mathrm {F} (1,\nu _{2})\,}

then

X
=

lim

ν

2

→
∞

Y

{\displaystyle X=\lim _{\nu _{2}\to \infty }Y\,}

has the chi-squared distribution

χ

1

2

{\displaystyle \chi _{1}^{2}}

‖

N

i
=
1
,
…
,
k

(
0
,
1
)

‖

2

∼

χ

k

2

{\displaystyle \|{\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
∼

χ

ν

2

{\displaystyle X\sim \chi _{\nu }^{2}\,}

and

c
>
0

{\displaystyle c>0\,}

, then

c
X
∼
Γ
(
k
=
ν

/

2
,
θ
=
2
c
)

{\displaystyle cX\sim \Gamma (k=\nu /2,\theta =2c)\,}

. (gamma distribution)
If

X
∼

χ

k

2

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

then

X

∼

χ

k

{\displaystyle {\sqrt {X}}\sim \chi _{k}}

(chi distribution)
If

X
∼

χ

2

2

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

, then

X
∼
Exp
⁡
(
1

/

2
)

{\displaystyle X\sim \operatorname {Exp} (1/2)}

is an exponential distribution. (See gamma distribution for more.)
If

X
∼

χ

2
k

2

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

, then

X
∼
Erlang
⁡
(
k
,
1

/

2
)

{\displaystyle X\sim \operatorname {Erlang} (k,1/2)}

is an Erlang distribution.
If

X
∼
Erlang
⁡
(
k
,
λ
)

{\displaystyle X\sim \operatorname {Erlang} (k,\lambda )}

, then

2
λ
X
∼

χ

2
k

2

{\displaystyle 2\lambda X\sim \chi _{2k}^{2}}

If

X
∼
Rayleigh
⁡
(
1
)

{\displaystyle X\sim \operatorname {Rayleigh} (1)\,}

(Rayleigh distribution) then

X

2

∼

χ

2

2

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

If

X
∼
Maxwell
⁡
(
1
)

{\displaystyle X\sim \operatorname {Maxwell} (1)\,}

(Maxwell distribution) then

X

2

∼

χ

3

2

{\displaystyle X^{2}\sim \chi _{3}^{2}\,}

If

X
∼

χ

ν

2

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

then

1
X

∼

I
n
v
-

⁡

χ

ν

2

{\displaystyle {\tfrac {1}{X}}\sim \operatorname {Inv-} \chi _{\nu }^{2}\,}

(Inverse-chi-squared distribution)
The chi-squared distribution is a special case of type III Pearson distribution
If

X
∼

χ

ν

1

2

{\displaystyle X\sim \chi _{\nu _{1}}^{2}\,}

and

Y
∼

χ

ν

2

2

{\displaystyle Y\sim \chi _{\nu _{2}}^{2}\,}

are independent then

X

X
+
Y

∼
Beta
⁡
(

ν

1

2

,

ν

2

2

)

{\displaystyle {\tfrac {X}{X+Y}}\sim \operatorname {Beta} ({\tfrac {\nu _{1}}{2}},{\tfrac {\nu _{2}}{2}})\,}

(beta distribution)
If

X
∼
U
⁡
(
0
,
1
)

{\displaystyle X\sim \operatorname {U} (0,1)\,}

(uniform distribution) then

−
2
log
⁡
(
X
)
∼

χ

2

2

{\displaystyle -2\log(X)\sim \chi _{2}^{2}\,}

If

X

i

∼
Laplace
⁡
(
μ
,
β
)

{\displaystyle X_{i}\sim \operatorname {Laplace} (\mu ,\beta )\,}

then

∑

i
=
1

n

2

|

X

i

−
μ

|

β

∼

χ

2
n

2

{\displaystyle \sum _{i=1}^{n}{\frac {2|X_{i}-\mu |}{\beta }}\sim \chi _{2n}^{2}\,}

If

X

i

{\displaystyle X_{i}}

follows the generalized normal distribution (version 1) with parameters

μ
,
α
,
β

{\displaystyle \mu ,\alpha ,\beta }

then

∑

i
=
1

n

2

|

X

i

−
μ

|

β

α

∼

χ

2
n

/

β

2

{\displaystyle \sum _{i=1}^{n}{\frac {2|X_{i}-\mu |^{\beta }}{\alpha }}\sim \chi _{2n/\beta }^{2}\,}

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

{\displaystyle k}

degrees of freedom is defined as the sum of the squares of

k

{\displaystyle k}

independent standard normal random variables.
If

Y

{\displaystyle Y}

is a

k

{\displaystyle k}

-dimensional Gaussian random vector with mean vector

μ

{\displaystyle \mu }

and rank

k

{\displaystyle k}

covariance matrix

C

{\displaystyle C}

, then

X
=
(
Y
−
μ

)

T

C

−
1

(
Y
−
μ
)

{\displaystyle X=(Y-\mu )^{T}C^{-1}(Y-\mu )}

is chi-squared distributed with

k

{\displaystyle 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

{\displaystyle Y}

is a vector of

k

{\displaystyle k}

i.i.d. standard normal random variables and

A

{\displaystyle A}

is a

k
×
k

{\displaystyle k\times k}

symmetric, idempotent matrix with rank

k
−
n

{\displaystyle k-n}

, then the quadratic form

Y

T

A
Y

{\displaystyle Y^{T}AY}

is chi-square distributed with

k
−
n

{\displaystyle 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

+
⋯
+

w

p

=
1

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

and

w

i

≥
0
,
i
=
1
,
…
,
p
,

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

then

1

(

w

1

X

1

,
…
,

w

p

X

p

)

Σ

(

w

1

X

1

,
…
,

w

p

X

p

)

⊤

∼

χ

1

2

.

{\displaystyle {\frac {1}{\left({\frac {w_{1}}{X_{1}}},\ldots ,{\frac {w_{p}}{X_{p}}}\right)\Sigma \left({\frac {w_{1}}{X_{1}}},\ldots ,{\frac {w_{p}}{X_{p}}}\right)^{\top }}}\sim \chi _{1}^{2}.}

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

Y

{\displaystyle Y}

is F-distributed,

Y
∼
F
(

k

1

,

k

2

)

{\displaystyle Y\sim F(k_{1},k_{2})}

if

Y
=

X

1

/

k

1

X

2

/

k

2

{\displaystyle Y={\frac {{X_{1}}/{k_{1}}}{{X_{2}}/{k_{2}}}}}

, where

X

1

∼

χ

k

1

2

{\displaystyle X_{1}\sim \chi _{k_{1}}^{2}}

and

X

2

∼

χ

k

2

2

{\displaystyle X_{2}\sim \chi _{k_{2}}^{2}}

are statistically independent.
If

X

1

∼

χ

k

1

2

{\displaystyle X_{1}\sim \chi _{k_{1}}^{2}}

and

X

2

∼

χ

k

2

2

{\displaystyle X_{2}\sim \chi _{k_{2}}^{2}}

are statistically independent, then

X

1

+

X

2

∼

χ

k

1

+

k

2

2

{\displaystyle X_{1}+X_{2}\sim \chi _{k_{1}+k_{2}}^{2}}

. If

X

1

{\displaystyle X_{1}}

and

X

2

{\displaystyle X_{2}}

are not independent, then

X

1

+

X

2

{\displaystyle X_{1}+X_{2}}

is not chi-square 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

,
…
,

X

n

{\displaystyle X_{1},\ldots ,X_{n}}

are chi square random variables and

a

1

,
…
,

a

n

∈

R

>
0

{\displaystyle a_{1},\ldots ,a_{n}\in \mathbb {R} _{>0}}

, then the distribution of

X
=

∑

i
=
1

n

a

i

X

i

{\displaystyle X=\sum _{i=1}^{n}a_{i}X_{i}}

is a special case of a Generalized Chi-squared Distribution.
A closed expression for this distribution is not known. It may be, however, approximated efficiently using the property of characteristic functions of chi-square random variables.

= Chi-squared distributions

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

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
∼

χ

k

2

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

is a special case of the gamma distribution, in that

X
∼
Γ

(

k
2

,

1
2

)

{\displaystyle X\sim \Gamma \left({\frac {k}{2}},{\frac {1}{2}}\right)}

using the rate parameterization of the gamma distribution (or

X
∼
Γ

(

k
2

,
2

)

{\displaystyle 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
∼

χ

2

2

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

, then

X
∼
Exp
⁡

(

1
2

)

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

is an exponential distribution.
The Erlang distribution is also a special case of the gamma distribution and thus we also have that if

X
∼

χ

k

2

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

with even

k

{\displaystyle k}

, then

X

{\displaystyle X}

is Erlang distributed with shape parameter

k

/

2

{\displaystyle k/2}

and scale parameter

1

/

2

{\displaystyle 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

X

1

,
.
.
.
,

X

n

{\displaystyle X_{1},...,X_{n}}

are i.i.d.

N
(
μ
,

σ

2

)

{\displaystyle N(\mu ,\sigma ^{2})}

random variables, then

∑

i
=
1

n

(

X

i

−

X

i

¯

)

2

∼

σ

2

χ

n
−
1

2

{\displaystyle \sum _{i=1}^{n}(X_{i}-{\overline {X_{i}}})^{2}\sim \sigma ^{2}\chi _{n-1}^{2}}

where

X

i

¯

=

1
n

∑

i
=
1

n

X

i

{\displaystyle {\overline {X_{i}}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}}

.
The box below shows some statistics based on

X

i

∼
N
(

μ

i

,

σ

i

2

)
,
i
=
1
,
…
,
k

{\displaystyle X_{i}\sim N(\mu _{i},\sigma _{i}^{2}),i=1,\ldots ,k}

independent random variables that have probability distributions related to the chi-squared distribution:

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

Computational methods

= Table of χ2 values vs p-values

=
The

p

{\textstyle 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 non-significant results.
The table below gives a number of p-values matching to

χ

2

{\displaystyle \chi ^{2}}

for the first 10 degrees of freedom.

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 2.1673 ≈ 2.17 as in the table above, noticing that 1 – p is the p-value from the table.

History

This distribution was first described by the German geodesist and 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-square" 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 −½xTΣ−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.

References

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
Values of the Chi-squared distribution

Definitions

= Introduction

= Probability density function

= Cumulative distribution function

Properties

= Cochran's theorem

= Additivity

= Sample mean

= Entropy

= Noncentral moments

= Cumulants

= Concentration

= Asymptotic properties

Related distributions

= Generalizations

= Linear combination

= Chi-squared distributions

= Gamma, exponential, and related distributions

Occurrence and applications

Computational methods

= Table of χ2 values vs p-values

History

See also

References

Further reading

External links

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