Chi-Square Distribution

(redirected from chi-square)
Also found in: Dictionary, Medical, Wikipedia.
Related to chi-square: chi-square 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.

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

References in periodicals archive ?
2] had developed to these stages (Stage 13; chi-square test, [chi square] = 30.
Familiarity with economic development organizations and growth aspirations--Pearson Chi-Square Tests Pearson Chi-Square Tests Familiarity Familiarity with GHP with NSBI Over the next 12 months Chi-square 0.
Because there were only four participants who stated they had more than 40 hours of IFRS training, the top two categories were combined for the chi-square analyses.
According to the information given in Table 1, Chi-square obtained for the teachers and professional at error probability level of 0.
We conducted chi-square and Cramer's V analyses per topic in relation to respondents' perceived ability to practice the 14 ASERVIC Competencies.
The chi-square contribution is displayed for each cell (in parentheses) to show how far the actual frequency differs from the expected frequency if the factors were independent.
Is it the one with the nominal chi-square, or, the step-wise analysis?
M2 EQUITYBITES-January 6, 2014-Cerenis Therapeutics reports primary endpoint results for Phase IIb CHI-SQUARE study
M2 PHARMA-January 6, 2014-Cerenis Therapeutics reports primary endpoint results for Phase IIb CHI-SQUARE study
The Pearson Chi-square test shows that there is no association between the two variables equality in provisions for skill development and gender as the value of significance is .