Chi-squared distribution

Chi-square pdf

Chi-squared distribution (χ²-distribution) is a widely used probability distribution in statistics. It is a special case of the gamma distribution and is one of the most used probability distributions in statistical hypothesis testing, particularly in chi-squared tests. The chi-squared distribution is used to test whether there is a significant difference between the expected and observed frequencies in one or more categories. It is also used in the construction of confidence intervals for a population standard deviation of a normal distribution from a sample standard deviation.

Definition

The chi-squared distribution with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. It is denoted by χ²(k), where k is the number of degrees of freedom (i.e., the number of independent standard normal variables in the sum).

Properties

The mean of the chi-squared distribution is equal to its degrees of freedom (k), and its variance is 2k. The distribution is skewed to the right, and its skewness decreases as the degrees of freedom increase. As k approaches infinity, the chi-squared distribution converges to a normal distribution.

Applications

The chi-squared distribution has numerous applications in statistical testing:

• In the chi-squared test for independence in contingency tables.
• In the chi-squared test for goodness of fit, to test whether sample data are consistent with a hypothesized distribution.
• In ANOVA (Analysis of Variance) for comparing more than two sample means.
• In the construction of confidence intervals for the variance of a normally distributed population.

Chi-squared Test

The chi-squared test uses the chi-squared distribution to evaluate hypotheses about the distribution of observed frequencies. The test statistic is calculated by comparing the observed frequencies with the expected frequencies under the null hypothesis. The resulting test statistic follows a chi-squared distribution if the null hypothesis is true.

See also

• Gamma distribution
• Normal distribution
• Statistical hypothesis testing
• Degrees of freedom (statistics)

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