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 ?
As shown in Table 3, the Chi-square test was used to verify the relationship between the PFGE pattern and sample type.
Further, separate chi-square tests revealed that boys were 1.
According to the information given in Table 1, Chi-square obtained for the teachers and professional at error probability level of 0.
The Pearson Chi-square test shows that there is no association between the two variables, ranking of employees on qualification basis and gender, as the value of significance is .
The density of normal BMI is higher among the subjects in the urban area (Pearson Chi-Square = 16.
For the hook-and-line study, a chi-square analysis was conducted to compare control and procedural control hook data in order to determine whether the presence of an object (sham-magnet) on a hook altered fish capture.
Tests of the association between dental erosion and factors were carried out using Mann-Whitney-U, unpaired t test, Chi-square test.
2]), being [chi square] distributed when summed across subjects and across time as the cumulative deviation of chi-square, were used as an index of deviation from random (Bierman, 1996).
In order to examine the relationship between the level of empowerment and the socio economic factors, the Chi-square test was used.