Skip to main content

Chi square distribution

Chi-Square Distribution

Probability Density FunctionThe chi-square distribution results when nu independent variables with standard normal distributions are squared and summed. The formula for the probability density function of the chi-square distribution isf(x) = EXP(-x/2)*x**(nu/2 - 1)/(2**(nu/2)*GAMMA(nu/2))  x >= 0
where nu is the shape parameter and GAMMA is the gamma function. The formula for the gamma function is
GAMMA(a) = INTEGRAL[t**(a-1)*EXP(-t)dt]  where the
 integration is from 0 to infinity
In a testing context, the chi-square distribution is treated as a "standardized distribution" (i.e., no location or scale parameters). However, in a distributional modeling context (as with other probability distributions), the chi-square distribution itself can be transformed with a location parametermu, and a scale parametersigma.
The following is the plot of the chi-square probability density function for 4 different values of the shape parameter.
plot of the chi-square probability density function for 4
 different values of the shape parameter
Cumulative Distribution FunctionThe formula for the cumulative distribution function of the chi-square distribution isF(x) = gamma(nu/2,x/2)/GAMMA(nu/2)   for x >= 0
where GAMMA is the gamma function defined above and gamma is the incomplete gamma function. The formula for the incomplete gamma function is
GAMMA(a,x) = INTEGRAL[t**(a-1)*EXP(-t)dt]  where the
 integration is from 0 to x
The following is the plot of the chi-square cumulative distribution function with the same values of  as the pdf plots above.
plot of the chi-square cumulative distribution function with
 the same values of nu as the pdf plots above
Percent Point FunctionThe formula for the percent point function of the chi-square distribution does not exist in a simple closed form. It is computed numerically.The following is the plot of the chi-square percent point function with the same values of nu as the pdf plots above.
plot of the chi-square percent point function with the same
 values of nu as the pdf plots above
Other Probability FunctionsSince the chi-square distribution is typically used to develop hypothesis tests and confidence intervals and rarely for modeling applications, we omit the formulas and plots for the hazard, cumulative hazard, survival, and inverse survival probability functions.
Common Statistics
Meannu
Medianapproximately nu - 2/3 for large nu
Modenu -2   for nu > 2
Range0 to positive infinity
Standard DeviationSQRT(2*nu)
Coefficient of VariationSQRT(2/nu)
Skewness2**1.5/SQRT(nu)
Kurtosis3 + 12/nu
Parameter EstimationSince the chi-square distribution is typically used to develop hypothesis tests and confidence intervals and rarely for modeling applications, we omit any discussion of parameter estimation.
CommentsThe chi-square distribution is used in many cases for the critical regions for hypothesis tests and in determining confidence intervals. Two common examples are the chi-square test for independence in an RxC contingency table and the chi-square test to determine if the standard deviation of a population is equal to a pre-specified value.
SoftwareMost general purpose statistical software programs support at least some of the probability functions for the chi-square distribution.

Comments

Popular posts from this blog

Frequency Polygons

Learning Objectives Create and interpret frequency polygons Create and interpret cumulative frequency polygons Create and interpret overlaid frequency polygons Frequency polygons are a graphical device for understanding the shapes of distributions. They serve the same purpose as histograms, but are especially helpful for comparing sets of data. Frequency polygons are also a good choice for displaying cumulative frequency distributions . To create a frequency polygon, start just as for histograms , by choosing a class interval. Then draw an X-axis representing the values of the scores in your data. Mark the middle of each class interval with a tick mark, and label it with the middle value represented by the class. Draw the Y-axis to indicate the frequency of each class. Place a point in the

Lognormal distribution

Lognormal Distribution Probability Density Function A variable X is lognormally distributed if Y = LN(X) is normally distributed with "LN" denoting the natural logarithm. The general formula for the  probability density function  of the lognormal distribution is where   is the  shape parameter ,   is the  location parameter  and  m is the  scale parameter . The case where   = 0 and  m  = 1 is called the  standard lognormal distribution . The case where   equals zero is called the 2-parameter lognormal distribution. The equation for the standard lognormal distribution is Since the general form of probability functions can be  expressed in terms of the standard distribution , all subsequent formulas in this section are given for the standard form of the function. The following is the plot of the lognormal probability density function for four values of  . There are several common parameterizations of the lognormal distribution. The form given here is from  Evans, Ha

Basics of Sampling Techniques

Population                A   population   is a group of individuals(or)aggregate of objects under study.It is also known as universe. The population is divided by (i)finite population  (ii)infinite population, (iii) hypothetical population,  subject to a statistical study . A population includes each element from the set of observations that can be made. (i) Finite population : A population is called finite if it is possible to count its individuals. It may also be called a countable population. The number of vehicles crossing a bridge every day, (ii) Infinite population : Sometimes it is not possible to count the units contained in the population. Such a population is called infinite or uncountable. ex, The number of germs in the body of a patient of malaria is perhaps something which is uncountable   (iii) Hypothetical population : Statistical population which has no real existence but is imagined to be generated by repetitions of events of a certain typ