Chi-Square Distribution

(redirected from Chi2)
Also found in: Dictionary, Medical.
Related to Chi2: chi2 test

chi-square distribution

[′kī ¦skwer dis·trə′byü·shən]
(statistics)
The distribution of the sum of the squares of a set of variables, each of which has a normal distribution and is expressed in standardized units.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Chi-Square Distribution

 

The probability distribution of the sum

of the squares of the normally distributed random variables X1, . . ., Xf, with zero mathematical expectation and unit variance is known as a chi-square distribution with f degrees of freedom. The distribution function for a chi-square random variable is

The first three moments (the mathematical expectation, the variance, and the third central moment) of χ2 are f, 2f, and 8f, respectively. The sum of two independent random variables Chi-Square Distribution and Chi-Square Distribution with f1 and f2 degrees of freedom has a chi-square distribution with f1 + f2 degrees of freedom.

Examples of chi-square distributions are the distributions of the squares of random variables that obey the Rayleigh and Maxwellian distributions. The Poisson distribution can be expressed in terms of a chi-square distribution with an even number of degrees of freedom:

If the number f of terms of the sum χ2 increases without bound, then, according to the central limit theorem, the distribution function of the standardized ratio Chi-Square Distribution converges to the standard normal distribution:

where

A consequence of this fact is another limit relation, which is convenient for calculating Ff(x) when f has large values:

In mathematical statistics, the chi-square distribution is used to construct interval estimates and statistical tests. Let Yi, . . ., Yn be random variables representing independent measurements of an unknown constant a. Suppose the measurement errors Yia are independent and are distributed identically normally. We have

E(Yia) = 0 E(Yia)2 = σ2

The statistical estimate of the unknown variance σ2 is then expressed by the equation

s2 = S2/(n – 1)

where

The ratio S22 obeys a chi-square distribution with f = n – 1 degrees of freedom. Let x1 and x2 be positive numbers that are solutions of the equations Ff(x1) = α/2 and Ff(x2) = 1 – α/2, where α is a specified number in the interval (0,1/2). In this case

P{x1 < S22 < x2} = P{S2/x2 < σ2 < S2/x1} = 1 – α

The interval (S2/x1, S2/x2 is called the confidence interval for σ2 with confidence coefficient 1 – α.

This method of constructing an interval estimate for σ2 is often used to test the hypothesis that Chi-Square Distribution, where Chi-Square Distribution is a given number. Thus, if Chi-Square Distribution belongs to the confidence interval indicated, then one concludes that the measurements do not contradict the hypothesis Chi-Square Distribution. If, however, Chi-Square Distribution or Chi-Square Distribution, then it must be assumed that Chi-Square Distribution or Chi-Square Distribution, respectively. This test corresponds to a significance level equal to α.

REFERENCE

Cramer, H. Matematicheskie metody statistiki, 2nd ed. Moscow, 1975. (Translated from English.)

L. N. BOLSHEV

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
En revanche le test du Chi2 est tres fortement significatif (p=.000) pour les objets << droite >> et << gauche >> : c'est ainsi que les sujets qui effectuent, a propos de la << droite >>, une sortie du champ en accumulant des associations spatiales le font aussi a propos de la << gauche >>, et se retrouvent donc massivement dans la classe correspondante.
Chisquare test was applied for measurement and estimated individual values of genotype GBJA S757 males and females for the two models to compare their fitness (Table 5).The values Chi2 % parameter for GBJA and S757 genotype female; male in the Gompertz model wereestimated 21.43; 29.03% and 5.88; 2.22% and same parameter were found in Logistic model 71.43; 62.90% and13.24; 13.33% respectively.
Sobre las frecuencias resultantes de esos cruces se calcula un indice de correlacion (Chi2).
Table 4 Results of probit model estimation Variable Probit regression Coefficient t-value age 10 -0.0525 *** -2.89 marr -0.0564 -1.42 edu1 0.3882 *** 4.96 edu2 0.4430 *** 5.69 town 0.0638 * 1.74 disabled 0.4159 *** 5.18 language 0.2430 *** 5.93 cons -1.6202 *** -16.44 Number of spells = 9,487 Log likelihood = -3,027.5333 LR chi2(13) = 127.55 Prob > chi2 = 0.000 Pseudo R2 = 0,0206 Source: own computations using Stata Statistical Software ***,**,* - significant at 1%, 5%, 10% level respectively Since the sort order of the data could affect the results when using nearest-neighbor matching on a propensity score estimated with categorical (non-continuous) variables (Abadie et al.
Gender-based comparison was done via Chi2 test for categorical variables and T test for continuous variables.
Several indicators were used to evaluate the goodness of fit (Chi2, Root Mean Squared Error of Approximation, Standardized Root Mean Squared Residual, Comparative Fit Index, and Non-Normed Fit Index).