The chi-square distribution is a probability distribution that arises from the sum of the squares of independent standard normal random variables. It is denoted as χ² (chi-square) and has a single parameter called degrees of freedom (df). The degrees of freedom determine the shape and spread of the distribution.
The chi-square distribution is often used in statistics to test the goodness of fit between observed and expected frequencies in a contingency table or to assess the independence of two variables. It is also employed in hypothesis testing to analyze categorical data. Additionally, it plays a vital role in various fields like genetics, insurance, and quality control.
The shape of the chi-square distribution is skewed to the right and its exact form depends on the number of degrees of freedom. As the degrees of freedom increase, the distribution becomes more symmetric and approaches a normal distribution. The mean of the chi-square distribution is equal to the degrees of freedom, and the variance is equal to twice the degrees of freedom.
The chi-square distribution is characterized by its probability density function (PDF), cumulative distribution function (CDF), and quantile function. These functions can be used to calculate probabilities, confidence intervals, and critical values associated with the chi-square distribution.
In summary, the chi-square distribution is a probability distribution used to analyze categorical data, test independence, and assess goodness of fit. It is defined by its degrees of freedom and has properties, such as skewness, mean, and variance, that depend on this parameter.
