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Probability distribution and special case of gamma distribution
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
Chi-squared_distribution
Noncentral generalization of the chi-squared distribution
the noncentral chi-squared distribution (or noncentral chi-square distribution, noncentral χ 2 {\displaystyle \chi ^{2}} distribution) is a noncentral
Noncentral chi-squared distribution
Noncentral_chi-squared_distribution
Probability distribution
variable and the origin. The chi distribution describes the positive square roots of a variable obeying a chi-squared distribution. If Z 1 , … , Z k {\displaystyle
Chi_distribution
Probability distribution
statistics, the inverse-chi-squared distribution (or inverted-chi-square distribution) is a continuous probability distribution of a positive-valued random
Inverse-chi-squared distribution
Inverse-chi-squared_distribution
Statistical hypothesis test
A chi-squared test (also chi-square or χ2 test) is a statistical hypothesis test used in the analysis of contingency tables when the sample sizes are
Chi-squared_test
Kind of probability distribution
and statistics, the generalized chi-squared distribution (or generalized chi-square distribution) is the distribution of a quadratic function of a multinormal
Generalized chi-squared distribution
Generalized_chi-squared_distribution
Probability distribution
of scaled inverse chi-squared distributions is linked to the inverse-chi-squared distribution and to the chi-squared distribution: If X ∼ ψ inv- χ 2
Scaled inverse chi-squared distribution
Scaled_inverse_chi-squared_distribution
Statistical measure of how far values spread from their average
theorem shows that the unbiased sample variance S2 follows a scaled chi-squared distribution (see also: asymptotic properties and an elementary proof): ( n
Variance
Evaluates how likely it is that any difference between data sets arose by chance
Pearson's chi-squared test or Pearson's χ 2 {\displaystyle \chi ^{2}} test is a statistical test applied to sets of categorical data to evaluate how likely
Pearson's_chi-squared_test
Topics referred to by the same term
The term chi-square, chi-squared, or χ 2 {\displaystyle \chi ^{2}} has various uses in statistics: chi-square distribution, a continuous probability distribution
Chi-square
Continuous probability distribution
since the chi-squared distribution is the sum of squares of independent standard normal random variables, the random variable of the F-distribution may also
F-distribution
Type of probability distribution
T-squared distribution (T2), proposed by Harold Hotelling, is a multivariate probability distribution that is tightly related to the F-distribution and
Hotelling's T-squared distribution
Hotelling's_T-squared_distribution
Statistical test used on paired nominal data
2 {\displaystyle \chi ^{2}} has a chi-squared distribution with 1 degree of freedom. If the χ 2 {\displaystyle \chi ^{2}} result is significant, this
McNemar's_test
Two-parameter family of continuous probability distributions
parametrize the inverse gamma distribution differently, as a scaled inverse chi-squared distribution. The inverse gamma distribution's probability density function
Inverse-gamma_distribution
characteristics related to the chi-squared distribution. Let random variable Y be defined as Y = X2 where X has normal distribution with mean 0 and variance
Proofs related to chi-squared distribution
Proofs_related_to_chi-squared_distribution
Gamma distribution, and it is used in goodness-of-fit tests in statistics. The inverse-chi-squared distribution The noncentral chi-squared distribution The
List of probability distributions
List_of_probability_distributions
Twenty-second letter of the Greek alphabet
various uses, including the chi-squared distribution, the chi-squared test, and chi-squared target model In algebraic topology, Chi is used to represent the
Chi_(letter)
Specific probability distribution function, important in physics
the above chi-squared distribution with one degree of freedom, and the total energy will be distributed according to a chi-squared distribution with five
Maxwell–Boltzmann distribution
Maxwell–Boltzmann_distribution
Probability distribution
{\textstyle |X-\mu |/\sigma \sim \chi _{1}} . The square of X / σ {\textstyle X/\sigma } has the noncentral chi-squared distribution with one degree of freedom:
Normal_distribution
Statistical theorem
\infty } , the distribution of the test statistic − 2 log ( Λ ) {\displaystyle -2\log(\Lambda )} asymptotically approaches the chi-squared ( χ 2 {\displaystyle
Wilks'_theorem
Topic in probability theory and statistics
gamma distribution with shape parameter α = v/2 and rate parameter β = 1/2 is a chi-squared distribution with ν degrees of freedom. A chi-squared distribution
Relationships among probability distributions
Relationships_among_probability_distributions
Probability distribution generalizing the F-distribution with a noncentrality parameter
F-distribution. It describes the distribution of the quotient (X/n1)/(Y/n2), where the numerator X has a noncentral chi-squared distribution with n1 degrees of freedom
Noncentral_F-distribution
Family of continuous probability distributions
the distribution simplifies to the chi-squared distribution with 2k degrees of freedom. It can therefore be regarded as a generalized chi-squared distribution
Erlang_distribution
Probability distribution
has a chi-squared distribution (χ2-distribution) with ν {\displaystyle \nu } degrees of freedom; Z and V are independent; A different distribution is defined
Student's_t-distribution
(central) chi-squared distribution is the distribution of a sum of squared independent standard normal distributions, i.e., normal distributions with mean
Noncentral_distribution
Probability distribution
distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution. There are two equivalent parameterizations
Gamma_distribution
Topics referred to by the same term
Mountain Chi-squared distribution, a theoretical probability distribution in inferential statistics Square (algebra), also known as the algebraic square Keyboard
X2
Generalization of gamma distribution to multiple dimensions
Wishart distribution. Chi-squared distribution Complex Wishart distribution F-distribution Gamma distribution Hotelling's T-squared distribution Inverse-Wishart
Wishart_distribution
Equation to approximate pooled degrees of freedom
probability distribution of χ' cannot be expressed analytically. However, its distribution can be approximated by another chi-squared distribution, whose effective
Welch–Satterthwaite_equation
Interference effect in radar systems
number of individual radiators and considering the result using the chi-squared distribution: p ( σ ) = m Γ ( m ) σ a v ( m σ σ a v ) m − 1 e − m σ σ a v I
Fluctuation_loss
noncentral chi distribution is a noncentral generalization of the chi distribution. It is also known as the generalized Rayleigh distribution. If X i {\displaystyle
Noncentral_chi_distribution
Probability distribution
^{2}}{2\sigma ^{2}}}} . Generate X {\displaystyle X} having a chi-squared distribution with 2 P + 2 {\displaystyle 2P+2} degrees of freedom. Set R
Rice_distribution
Statistical test
LM = n R 2 . {\displaystyle {\text{LM}}=nR^{2}.} This follows a chi-squared distribution, with degrees of freedom equal to P − 1, where P is the number
White_test
Discrete probability distribution
Poisson distribution can be expressed using the relationship between the cumulative distribution functions of the Poisson and chi-squared distributions. The
Poisson_distribution
Normality test
variance. If the data comes from a normal distribution, the JB statistic asymptotically has a chi-squared distribution with two degrees of freedom, so the statistic
Jarque–Bera_test
Test statistic
^{2}}{\nu }},} where the chi-squared is a weighted sum of squared deviations: χ 2 = ∑ i ( O i − C i ) 2 σ i 2 {\displaystyle \chi ^{2}=\sum _{i}{\frac
Reduced_chi-squared_statistic
Probability distribution
Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution
Rayleigh_distribution
Statistical method
fact can be used to determine the p-value for X2. The distribution of X2 is a chi-squared distribution for the following reason; under the null hypothesis
Fisher's_method
Generalization of the one-dimensional normal distribution to higher dimensions
p ) {\displaystyle \chi _{k}^{2}(p)} is the quantile function for probability p {\displaystyle p} of the chi-squared distribution with k {\displaystyle
Multivariate normal distribution
Multivariate_normal_distribution
Statistical method
correction modifies the standard chi-squared test to account for the fact that a continuous distribution (chi-squared) is used to approximate discrete
Yates's correction for continuity
Yates's_correction_for_continuity
Number of values in the final calculation of a statistic that are free to vary
with the squared lengths (or "sum of squares" of the coordinates) of such vectors, and the parameters of chi-squared and other distributions that arise
Degrees of freedom (statistics)
Degrees_of_freedom_(statistics)
Statistical measure of the discrepancy between data and an estimation model
sum of squares (RSS), also known as the sum of squared residuals (SSR) or the sum of squared estimate of errors (SSE), is the sum of the squares of residuals
Residual_sum_of_squares
Vector in statistics
addition, a quadratic form such as this follows a generalized chi-squared distribution. The case for general Λ {\displaystyle \Lambda } can be derived
Quadratic_form_(statistics)
population distribution if a specified test statistic is too large, when that statistic would have approximately a chi-square distribution if the null
Minimum_chi-square_estimation
Difficulties arising when analyzing data with many aspects ("dimensions")
. This also helps to understand the chi-squared distribution. Indeed, the (non-central) chi-squared distribution associated to a random point in the interval
Curse_of_dimensionality
Probability distribution
{\displaystyle \lambda >0} . χ n 2 {\displaystyle \chi _{n}^{2}} for the chi-squared distribution with n {\displaystyle n} degrees of freedom. P a r e
Exponential_distribution
Probability distribution
follows a normal distribution, and that both S 1 2 {\displaystyle S_{1}^{2}} and S 2 2 {\displaystyle S_{2}^{2}} has a chi-squared distribution, which is approximately
Log-normal_distribution
Value in statistics
of squares due to pure error, divided by the error variance σ2, has a chi-squared distribution with N − n degrees of freedom; The sum of squares due
Lack-of-fit_sum_of_squares
Normally distributed deviate
other marginal distributions may involve manipulating sequences of standard normal deviates: an example here is the chi-squared distribution, random values
Standard_normal_deviate
Statistical distribution of complex random variables
distribution (a complex normal distribution is a bivariate normal distribution) Generalized chi-squared distribution Wishart distribution Complex random variable
Complex_normal_distribution
Distribution of new data marginalized over the posterior
prior scaled-inverse-chi-squared distribution placed on σx2, with hyperparameters ν and σ2. The resulting compound distribution t ( x | μ , ν , σ 2 )
Posterior predictive distribution
Posterior_predictive_distribution
Function of the observed sample results
his Pearson's chi-squared test, using the chi-squared distribution and notated as capital P. The p-values for the chi-squared distribution (for various
P-value
Measure of inequality of a statistical distribution
November 2022. "Chi-Squared Distribution -- from Wolfram MathWorld". mathworld.wolfram.com. Retrieved 11 January 2023. "Weibull Distribution: Characteristics
Gini_coefficient
Generalization of the binomial distribution
converges in distribution to the chi-squared distribution χ 2 ( k − 1 ) {\displaystyle \chi ^{2}(k-1)} . [Proof] The space of all distributions over categories
Multinomial_distribution
non-central chi-squared distribution with zero degrees of freedom can be used in testing the null hypothesis that a sample is from a uniform distribution on the
Zero_degrees_of_freedom
Statistical test
G} is asymptotically chi-squared distributed as the total number of observations tends to infinity (convergence in distribution). Furthermore, the total
G-test
Probability distribution
by an independent chi-distributed random variable, while the F-distribution originates from the ratio of two independent chi-squared distributed random
Ratio_distribution
Probability distribution
{\displaystyle 2}{b}}\sum _{i=1}^{n}|X_{i}-\mu |\sim \chi ^{2}(2n)} (chi-squared distribution). If X , Y ∼ Laplace ( μ , b ) {\displaystyle X,Y\sim {\textrm
Laplace_distribution
Statistic used in signal detection theory
0}^{2}>0\right)} ( χ ~ 2 {\displaystyle {\tilde {\chi }}^{2}} denotes the generalized chi-squared distribution), where w = [ σ s 2 − σ n 2 ] , k = [ 1 1 ]
Sensitivity_index
Ways of computing statistical significance
One-tailed tests are used for asymmetric distributions that have a single tail, such as the chi-squared distribution, which are common in measuring goodness-of-fit
One-_and_two-tailed_tests
Probability distribution used in multivariate hypothesis testing
2(m-p+1)}.} Chi-squared distribution Dirichlet distribution F-distribution Gamma distribution Hotelling's T-squared distribution Student's t-distribution Wishart
Wilks's_lambda_distribution
Probability distribution
it is apparent from the fact that for the chi-squared distribution the excess kurtosis is 12/k and the square of the skewness is 8/k, hence (excess kurtosis
Beta_distribution
Statistical test of equal group variances
Cauchy distribution (a heavy-tailed distribution) and the median performed best when the underlying data followed a chi-squared distribution with four
Levene's_test
inequality Chernoff face Chernoff's distribution Chernoff's inequality Chi distribution Chi-squared distribution Chi-squared test Chinese restaurant process
List_of_statistics_articles
Metric for fit of statistical models
determination (the R-squared measure of goodness of fit); Lack-of-fit sum of squares; Mallows's Cp criterion Prediction error Reduced chi-square The following
Goodness_of_fit
Probability distribution
{\displaystyle X\,\sim \,\operatorname {Scale-inv-\chi ^{2}} (1,c)} (scaled-inverse-chi-squared distribution). If X ∼ Levy ( μ , c ) {\displaystyle X\sim
Lévy_distribution
Statistical model for a binary dependent variable
}}_{\varphi })=11.6661\ldots } Using the chi-squared test of significance, the integral of the chi-squared distribution with one degree of freedom from 11.6661
Logistic_regression
the contingency table of the variable set. The test inherits a chi-squared distribution with data-set adaptive degrees of freedom f, estimated from smaller
W-test
Statistical transform
\,} Because R2 is the square of the norm of the standard bivariate normal variable (X, Y), it has the chi-squared distribution with two degrees of freedom
Box–Muller_transform
~N~} increases, the distribution of − 2 ln ( [ L R ] ) {\displaystyle ~-2\ln([{\mathcal {LR}}])~} converges to that of chi-squared with k − 1 {\displaystyle
Multinomial_test
Statistics concept
r_{i}=X_{i}-{\overline {X}}.} The sum of squares of the statistical errors, divided by σ2, has a chi-squared distribution with n degrees of freedom: 1 σ 2 ∑
Errors_and_residuals
Observation that in many real-life datasets, the leading digit is likely to be small
F-distribution is fitted well for low degrees of freedom. With increasing dfs the fit decreases but much more slowly than the chi-squared distribution.
Benford's_law
German geodesist
In 1876 he discovered the chi-squared distribution as the distribution of the sample variance for a normal distribution. This discovery and other of
Friedrich_Robert_Helmert
Statistical distance measure
dimensions other than 2, the cumulative chi-squared distribution should be consulted. In a normal distribution, the region where the Mahalanobis distance
Mahalanobis_distance
Principle in genetics
classes. If this is the case, then the asymptotic assumption of the chi-squared distribution, will no longer hold, and it may be necessary to use a form of
Hardy–Weinberg_principle
Family of continuous probability distributions
{Gamma} (m+1,b_{1}^{2})} . The Pearson type III distribution is a gamma distribution or chi-squared distribution. Defining new parameters: C 1 = b 1 2 b 2
Pearson_distribution
Statistical distribution
special case of the generalized chi-squared distribution. A related concept is the generalized near-integer gamma distribution (GNIG). The random variable
Generalized integer gamma distribution
Generalized_integer_gamma_distribution
Statistical test
true, then the probability distribution of the test statistic can be made as close as desired to the chi-squared distribution with 5 degrees of freedom
Exact_test
Signal processing phenomenon
by generating a random variable from the appropriate noncentral chi-squared distribution. The Heston model is a stochastic volatility model used in mathematical
Multiplicative_noise
278. ISBN 978-1-420-01137-1. Notes More precisely 1.53817..., the square root of the median of a chi-squared distribution with 3 degrees of freedom.
Misconceptions about the normal distribution
Misconceptions_about_the_normal_distribution
Method for estimating the unknown parameters in a linear regression model
proportional to the chi-squared distribution: s 2 ∼ σ 2 n − p ⋅ χ n − p 2 {\displaystyle s^{2}\ \sim \ {\frac {\sigma ^{2}}{n-p}}\cdot \chi _{n-p}^{2}} The
Ordinary_least_squares
Statistical theorem in the analysis of variance
r_{1}+\cdots +r_{k}=N} , the Qi are independent each Qi has a chi-squared distribution with ri degrees of freedom. Often it's stated as ∑ i A i = A {\displaystyle
Cochran's_theorem
Statistical significance test
the sampling distribution of the test statistic that is calculated is only approximately equal to the theoretical chi-squared distribution. The approximation
Fisher's_exact_test
Statistical test for logistic regression models
the p-value Compare the computed Hosmer–Lemeshow statistic to a chi-squared distribution with Q − 2 degrees of freedom to calculate the p-value. There are
Hosmer–Lemeshow_test
Analog graphical calculator
values needed when performing a familiar statistical test, Pearson's chi-squared test. This nomogram demonstrates the use of curved scales with unevenly
Nomogram
Topics referred to by the same term
Box-Pierce test outputs a Q-statistic (uppercase) which follows the chi-squared distribution The Ljung-Box test is a modified version of the Box-Pierce test
Q-statistic
Decision tree learning technique
that it is non-parametric.[citation needed] Bonferroni correction Chi-squared distribution Decision tree learning Latent class model Market segment Multiple
Chi-square automatic interaction detection
Chi-square_automatic_interaction_detection
Mathematical function for the probability a given outcome occurs in an experiment
chi-squared test) Student's t distribution, the distribution of the ratio of a standard normal variable and the square root of a scaled chi squared variable;
Probability_distribution
Branch of statistics
chi-squared test) Student's t distribution, the distribution of the ratio of a standard normal variable and the square root of a scaled chi squared variable;
Mathematical_statistics
Number useful in statistics for analyzing a normal curve
975}}e^{-x^{2}/2}\,\mathrm {d} x=0.95.} Its square, about 3.84146, is the 95th percentile point of a chi-squared distribution with 1 degree of freedom, often used
97.5th_percentile_point
Property of having a unique mode or maximum value
normal distributions, which are unimodal. Other examples of unimodal distributions include Cauchy distribution, Student's t-distribution, chi-squared distribution
Unimodality
Non-parametric method for testing whether samples originate from the same distribution
distribution of H {\displaystyle H} can be quite different from this chi-squared distribution. If a table of the chi-squared probability distribution
Kruskal–Wallis_test
with a multivariate Gaussian joint distribution, the square of the Mahalanobis distance has a chi-squared distribution with n {\displaystyle n} degrees
Landing_footprint
Probability theory
inverse distributions include inverse-chi-squared distribution inverse-gamma distribution inverse-Wishart distribution inverse matrix gamma distribution Inverse
Inverse_distribution
Statistical property
for a chi-squared distribution with the degrees of freedom equal to the number of independent variables. The null hypothesis of this chi-squared test is
Homoscedasticity and heteroscedasticity
Homoscedasticity_and_heteroscedasticity
Stochastic model for the evolution of financial interest rates
c={\frac {2a}{(1-e^{-aT})\sigma ^{2}}}} , and Y is a non-central chi-squared distribution with 4 a b σ 2 {\displaystyle {\frac {4ab}{\sigma ^{2}}}} degrees
Cox–Ingersoll–Ross_model
Statistical test
− 1 2 {\displaystyle T>\chi _{1-\alpha ,k-1}^{2}} where Χ21 − α,k − 1 is the (1 − α)-quantile of the chi-squared distribution with k − 1 degrees of freedom
Cochran's_Q_test
Topics referred to by the same term
"Pearson's r" Pearson's chi-squared test, a statistical procedure whose results are evaluated by reference to the chi-square distribution Pearson's Chapel,
Pearson's
Probabilistic classification algorithm
approach.* Gary Robinson's Linux Journal article discussed using the chi squared distribution. "Archived copy". Archived from the original on 2016-10-07. Retrieved
Naive_Bayes_classifier
Statistical property
(T^{2})\end{aligned}}} If the variables X1 ... Xn follow a normal distribution, then nS2/σ2 has a chi-squared distribution with n − 1 degrees of freedom, giving: E [ n
Bias_of_an_estimator
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CHI SQUARED-DISTRIBUTION
CHI SQUARED-DISTRIBUTION
CHI SQUARED-DISTRIBUTION
CHI SQUARED-DISTRIBUTION
CHI SQUARED-DISTRIBUTION
CHI SQUARED-DISTRIBUTION
CHI SQUARED-DISTRIBUTION
CHI SQUARED-DISTRIBUTION
CHI SQUARED-DISTRIBUTION
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