Chi Square Distribution

Information about Chi Square Distribution

This article is about the mathematics of the chi-square distribution. For its uses in statistics, see chi-square test.

chi-square
Probability density function
Cumulative distribution function
Parameters	$\text{[math]}$ degrees of freedom
Support	$\text{[math]}$
Probability density function (pdf)
Cumulative distribution function (cdf)
Mean	$\text{[math]}$
Median	approximately
Mode	if
Variance
Skewness
Excess kurtosis
Entropy
Moment-generating function (mgf)	for
Characteristic function

In probability theory and statistics, the chi-square distribution (also chi-squared or $\text{[math]}$ distribution) is one of the most widely used theoretical probability distributions in inferential statistics, i.e. in statistical significance tests. It is useful because, under reasonable assumptions, easily calculated quantities can be proven to have distributions that approximate to the chi-square distribution if the null hypothesis is true.

If $\text{[math]}$ are k independent, normally distributed random variables with mean 0 and variance 1, then the random variable

is distributed according to the chi-square distribution. This is usually written

The chi-square distribution has one parameter: $\text{[math]}$ - a positive integer that specifies the number of degrees of freedom (i.e. the number of $\text{[math]}$ )

The chi-square distribution is a special case of the gamma distribution.

The best-known situations in which the chi-square distribution is used are the common chi-square tests for goodness of fit of an observed distribution to a theoretical one, and of the independence of two criteria of classification of qualitative data. However, many other statistical tests lead to a use of this distribution. One example is Friedman's analysis of variance by ranks.

Characteristics

Probability density function

A probability density function of the chi-square distribution is

where $\text{[math]}$ denotes the Gamma function, which takes particular values at the half-integers.

Cumulative distribution function

Its cumulative distribution function is:

where $\text{[math]}$ is the lower incomplete Gamma function and is the regularized Gamma function.

Tables of this distribution — usually in its cumulative form — are widely available and the function is included in many spreadsheets and all statistical packages.

Characteristic function

The characteristic function of the Chi-square distribution is

Properties

The chi-square distribution has numerous applications in inferential statistics, for instance in chi-square 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 divided by their respective degrees of freedom.

Normal approximation

If , then as $\text{[math]}$ tends to infinity, the distribution of $\text{[math]}$ tends to normality. However, the tendency is slow (the skewness is and the kurtosis excess is ) and two transformations are commonly considered, each of which approaches normality faster than $\text{[math]}$ itself:

Fisher emprically showed that is approximately normally distributed with mean and unit variance. It is possible to arrive at the same normal approximation result by using moment matching. To see this, consider the mean and the variance of a Chi-distributed random variable , which are given by and , where $\text{[math]}$ is the Gamma function. The particular ratio of the Gamma functions in has the following series expansion [1]: When , this ratio can be approximated as follows:

Then, simple moment matching results in the following approximation of $\text{[math]}$ : , from which it follows that .

Wilson and Hilferty showed in 1931 that is approximately normally distributed with mean and variance .

The expected value of a random variable having chi-square distribution with $\text{[math]}$ degrees of freedom is $\text{[math]}$ and the variance is $\text{[math]}$ . The median is given approximately by

Note that 2 degrees of freedom lead to an exponential distribution.

Information entropy

The information entropy is given by

where $\text{[math]}$ is the Digamma function.

Related distributions

is an exponential distribution if $\text{[math]}$ (with 2 degrees of freedom).
is a chi-square distribution if for independent that are normally distributed.
If the have nonzero means, then is drawn from a noncentral chi-square distribution.
The chi-square distribution is a special case of the gamma distribution, in that .
is an F-distribution if where and are independent with their respective degrees of freedom.
is a chi-square distribution if where are independent and .
if $\text{[math]}$ is chi-square distributed, then is chi distributed.
in particular, if $\text{[math]}$ (chi-square with 2 degrees of freedom), then is Rayleigh distributed.
if $\text{[math]}$ are i.i.d. random variables, then where .
if , then

**Various chi and chi-square distributions**
Name	Statistic
chi-square distribution
noncentral chi-square distribution
chi distribution
noncentral chi distribution

External links

On-line calculator for the significance of chi-square, in Richard Lowry's statistical website at Vassar College.
Distribution Calculator Calculates probabilities and critical values for normal, t-, chi2- and F-distribution
Chi-Square Calculator for critical values of Chi-Square in R. Webster West's applet website at University of South Carolina
Chi-Square Calculator from GraphPad

	Probability distributions [ • • [ edit] ]
	Univariate	Multivariate
Discrete:	Benford • Bernoulli • binomial • Boltzmann • categorical • compound Poisson • discrete phase-type • degenerate • Gauss-Kuzmin • geometric • hypergeometric • logarithmic • negative binomial • parabolic fractal • Poisson • Rademacher • Skellam • uniform • Yule-Simon • zeta • Zipf • Zipf-Mandelbrot	Ewens • multinomial • multivariate Polya
Continuous:	Beta • Beta prime • Cauchy • chi-square • Dirac delta function • Coxian • Erlang • exponential • exponential power • F • fading • Fermi-Dirac • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Half-Logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square (scaled inverse chi-square) • inverse Gaussian • inverse gamma (scaled inverse gamma) • Kumaraswamy • Landau • Laplace • Lvy • Lvy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • Nakagami • normal (Gaussian) • normal-gamma • normal inverse Gaussian • Pareto • Pearson • phase-type • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • shifted Gompertz • Student's t • triangular • truncated normal • type-1 Gumbel • type-2 Gumbel • uniform • Variance-Gamma • Voigt • von Mises • Weibull • Wigner semicircle • Wilks' lambda	Dirichlet • Generalized Dirichlet distribution . inverse-Wishart • Kent • matrix normal • multivariate normal • multivariate Student • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous:	bimodal • Cantor • conditional • equilibrium • exponential family • infinitely divisible • location-scale family • marginal • maximum entropy • posterior • prior • quasi • sampling • singular

A chi-square test is any statistical hypothesis test in which the test statistic has a chi-square distribution when the null hypothesis is true, or any in which the probability distribution of the test statistic (assuming the null hypothesis is true) can be made to approximate a
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In mathematics, a support of a function f from a set X to the real numbers R is a subset Y of X such that f (x) is zero for all x in X and outside Y.
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In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals.

Formally, a probability distribution has density f, if f
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In probability theory, the cumulative distribution function (CDF), also called probability distribution function or just distribution function,^[1] completely describes the probability distribution of a real-valued random variable X.
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expected value (or mathematical expectation, or mean) of a discrete random variable is the sum of the probability of each possible outcome of the experiment multiplied by the outcome value (or payoff).
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median is described as the number separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking
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In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. The term is applied both to probability distributions and to collections of experimental data.
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variance of a random variable (or somewhat more precisely, of a probability distribution) is one measure of statistical dispersion, averaging the squared distance of its possible values from the expected value.
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skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable.

Introduction

Consider the distribution in the figure. The bars on the right side of the distribution taper differently than the bars on the left side.
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kurtosis (from the Greek word kurtos, meaning bulging) is a measure of the "peakedness" of the probability distribution of a real-valued random variable. Higher kurtosis means more of the variance is due to infrequent extreme deviations, as opposed to frequent
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Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable.

Shannon entropy quantifies the information contained in a piece of data: it is the minimum average message length, in bits (if using base-2 logarithms), that must
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In probability theory and statistics, the moment-generating function of a random variable X is

wherever this expectation exists. The moment-generating function generates the moments of the probability distribution.
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In probability theory, the characteristic function of any random variable completely defines its probability distribution. On the real line it is given by the following formula, where X is any random variable with the distribution in question:

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Probability theory is the branch of mathematics concerned with analysis of random phenomena.^[1] The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities
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Statistics is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It is applicable to a wide variety of academic disciplines, from the physical and social sciences to the humanities.
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The word theory has a number of distinct meanings in different fields of knowledge, depending on their methodologies and the context of discussion.

In common usage, people often use the word theory to signify a conjecture, an opinion, or a speculation.
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probability distribution that assigns a probability to every subset (more precisely every measurable subset) of its state space in such a way that the probability axioms are satisfied.
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Inferential statistics or statistical induction comprises the use of statistics to make inferences concerning some unknown aspect of a population. It is distinguished from descriptive statistics.
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In statistics, a result is called significant if it is unlikely to have occurred by chance. "A statistically significant difference" simply means there is statistical evidence that there is a difference; it does not mean the difference is necessarily large, important or significant
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In statistics, a null hypothesis is a hypothesis set up to be nullified or refuted in order to support an alternate hypothesis. When used, the null hypothesis is presumed true until statistical evidence in the form of a hypothesis test indicates otherwise.
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In probability theory, to say that two events are independent, intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs.
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normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. Each member of the family may be defined by two parameters, location and scale: the mean ("average",
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In statistics, mean has two related meanings:

the arithmetic mean (and is distinguished from the geometric mean or harmonic mean).
the expected value of a random variable, which is also called the population mean.

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This article or section is in need of attention from an expert on the subject.
Please help recruit one or [ improve this article] yourself. See the talk page for details.
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Goodness of fit means how well a statistical model fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measures can be used in statistical hypothesis testing, e.
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The Friedman test is a non-parametric statistical test developed by the U.S. economist Milton Friedman. Similar to the parametric repeated measures ANOVA, it is used to detect differences in treatments across multiple test attempts.
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Chi Square Distribution

Information about Chi Square Distribution

Characteristics

Probability density function

Cumulative distribution function

Characteristic function

Properties

Normal approximation

Information entropy

Related distributions

See also

External links

Introduction