The chi-square test is a statistical test used to determine whether there is a significant association between two categorical variables. It’s commonly used to analyze data that can be represented in a contingency table (also known as a cross-tabulation or crosstab).

Here’s how the chi-square test works:

1. Formulate Hypotheses:
   • Null Hypothesis (H0): There is no association between the two categorical variables.
   • Alternative Hypothesis (Ha): There is an association between the two categorical variables.
2. Set Significance Level (α): Typically, a significance level of 0.05 (5%) is chosen, but it can vary depending on the specific context and the desired level of confidence.
3. Construct Contingency Table: The data is organized into a contingency table, which is a tabular arrangement of the frequencies or counts of observations for each combination of categories of the two variables.
4. Calculate Expected Frequencies: Calculate the expected frequencies for each cell in the contingency table under the assumption that there is no association between the variables. This is done by multiplying the row total and column total for each cell and dividing by the overall total.
5. Calculate the Chi-Square Statistic: The chi-square statistic is calculated as the sum of the squared differences between the observed and expected frequencies, divided by the expected frequencies for all cells in the contingency table.
6. Determine the Degrees of Freedom: The degrees of freedom (df) for the chi-square test is calculated as (r – 1) * (c – 1), where r is the number of rows and c is the number of columns in the contingency table.
7. Look up Critical Value or Calculate P-value: Using the chi-square distribution with the calculated degrees of freedom, find the critical value corresponding to the chosen significance level (α), or calculate the p-value associated with the chi-square statistic.
8. Make a Decision: Compare the calculated chi-square statistic to the critical value from the chi-square distribution or compare the p-value to the chosen significance level. If the calculated chi-square statistic is greater than the critical value or if the p-value is less than the significance level (α), reject the null hypothesis and conclude that there is a significant association between the two variables.
9. Interpret Results: If the null hypothesis is rejected, interpret the results in the context of the specific research question or practical implications.

The chi-square test is commonly used in fields such as social sciences, market research, biology, and epidemiology to analyze categorical data and determine whether observed differences are statistically significant. It’s important to note that the chi-square test assumes that the observations are independent and that the expected frequencies in each cell of the contingency table are sufficiently large (typically at least 5). If these assumptions are violated, alternative tests or adjustments may be necessary.