Chi-Square Distribution

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, . . ., X_f, 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 and 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 converges to the standard normal distribution:

where

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

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

E(Y_i – a) = 0 E(Y_i – a)² = σ²

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

s² = S²/(n – 1)

where

The ratio S²/σ² 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 F_f(x₁) = α/2 and F_f(x₂) = 1 – α/2, where α is a specified number in the interval (0,1/2). In this case

P{x₁ < S²/σ² < 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 , where is a given number. Thus, if belongs to the confidence interval indicated, then one concludes that the measurements do not contradict the hypothesis . If, however, or , then it must be assumed that or , 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. BOL’SHEV

单词	chi-square distribution
释义	chi-square distribution chi-square distribution (ˈkaɪˌskwɛə) n (Statistics) statistics a continuous single-parameter distribution derived as a special case of the gamma distribution and used esp to measure goodness of fit and to test hypotheses and obtain confidence intervals for the variance of a normally distributed variable Chi-Square Distribution 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, . . ., X_f, 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 and 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 converges to the standard normal distribution: where A consequence of this fact is another limit relation, which is convenient for calculating F_f(x) when f has large values: In mathematical statistics, the chi-square distribution is used to construct interval estimates and statistical tests. Let Y_i, . . ., Y_n be random variables representing independent measurements of an unknown constant a. Suppose the measurement errors Y_i – a are independent and are distributed identically normally. We have E(Y_i – a) = 0 E(Y_i – a)² = σ² The statistical estimate of the unknown variance σ² is then expressed by the equation s² = S²/(n – 1) where The ratio S²/σ² 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 F_f(x₁) = α/2 and F_f(x₂) = 1 – α/2, where α is a specified number in the interval (0,1/2). In this case P{x₁ < S²/σ² < 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 , where is a given number. Thus, if belongs to the confidence interval indicated, then one concludes that the measurements do not contradict the hypothesis . If, however, or , then it must be assumed that or , 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. BOL’SHEV chi-square distribution chi-square dis·tri·bu·tion a variable is said to have a chi-square distribution with K degrees of freedom if it is distributed like the sum of the squares of K independent random variables, each of which has a normal (gaussian) distribution with mean zero and variance one. The chi-square distribution is the basis for many variations of the chi-square(d) test, perhaps the most widely used test for statistical significance in biology and medicine. LegalSeedistributionFinancialSeeDistribution
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chi-square distribution