Chikvadratfordelinger

Chikvadratfordelinger, also known as the chi-square distribution, is a continuous probability distribution that arises in the context of hypothesis testing and the analysis of categorical data. It is defined by the chi-square statistic, which is the sum of the squared differences between observed and expected frequencies in one or more categories, divided by the expected frequencies.

The chi-square distribution is characterized by a single parameter, k, which represents the degrees of freedom. The degrees of freedom are determined by the number of categories and any constraints imposed on the data. As k increases, the chi-square distribution approaches a normal distribution.

The chi-square distribution is widely used in statistical hypothesis testing, particularly in the chi-square test for independence and the chi-square goodness-of-fit test. These tests are used to determine whether there is a significant association between categorical variables or whether the observed data fits a hypothesized distribution.

The probability density function of the chi-square distribution is given by:

f(x; k) = (1 / (2^(k/2) * Gamma(k/2))) * x^((k/2) - 1) * e^(-x/2)

where x is a non-negative real number, k is the degrees of freedom, Gamma is the gamma function, and e is the base of the natural logarithm.

The cumulative distribution function of the chi-square distribution can be computed using the regularized gamma function or by numerical integration of the probability density function.

In summary, the chi-square distribution is a fundamental concept in statistics, used for hypothesis testing and the analysis of categorical data. It is characterized by a single parameter, the degrees of freedom, and has a wide range of applications in various fields.