The chi-square distribution is a probability distribution that is used in statistical analysis to test hypotheses about the variance or distribution of a population based on a sample. The chi-square distribution is used to determine whether the observed values of a categorical variable differ significantly from the expected values, and it is commonly used in the analysis of contingency tables.

The chi-square distribution is
characterized by a single parameter known as the degrees of freedom (df), which
is determined by the sample size and the number of variables being analysed. As
the degrees of freedom increase, the chi-square distribution approaches a normal
distribution, which is symmetrical and bell-shaped.

The chi-square test is a
statistical test that uses the chi-square distribution to test the null hypothesis
that there is no significant difference between the observed and expected
values of a categorical variable. The test involves calculating the chi-square
statistic, which is a measure of the difference between the observed and
expected values, and comparing it to a critical value obtained from the
chi-square distribution.

If the chi-square statistic is
greater than the critical value, the null hypothesis is rejected, and it is
concluded that there is a significant difference between the observed and
expected values. Conversely, if the chi-square statistic is less than the critical
value, the null hypothesis is not rejected.