Chi Squared Tests

The chi-squared test is a statistical method used to determine if there is a significant difference between observed and expected frequencies.
The test utilises the chi-squared distribution, which is part of a family of theoretical probability distributions that takes only positive values and is skewed to the right.
The null hypothesis for chi-squared tests assumes that there is no significant difference between the observed and expected frequencies.
The chi-squared statistic is calculated by summing the squared differences between observed and expected frequencies, divided by the expected frequency: X² = Σ[(O-E)²/E].
The degree of freedom for the chi-squared test is determined by the number of categories minus one: df = n - 1.
To compare the chi-squared statistic against a critical value, a chi-squared distribution table is used.
For a given significance level α (typically 0.05), if the calculated chi-squared statistic is greater than the critical value, the null hypothesis is rejected.
On the contrary, if the chi-squared statistic is less than or equal to the critical value, the null hypothesis is retained.
A chi-squared test can be used for either goodness of fit tests or tests for independence. A goodness of fit test checks if the observed sample distribution is consistent with a theoretical distribution, while tests for independence checks if events are independent.
One assumption of the chi-squared test is that observations are randomly sampled and independent of each other. Another is that the expected frequency for each category should be at least 5 for the approximation to be valid.
The chi-squared test is a non-parametric test, meaning it does not assume any underlying population distribution.
It’s important to remember that the chi-squared test only indicates if there is a significant difference, but not where the difference is.
Note that chi-squared tests can only compare data in the form of frequency counts and not actual numerical scores.
Misinterpretation or misuse of chi-squared test results may lead to type I or type II errors. These represent rejecting the null hypothesis when it is true (Type I) and failing to reject the null hypothesis when it is false (Type II).