The Chi-Square (pronounced 'kai-squared') test is a statistical hypothesis test used to determine if there is a significant association between two categorical variables.

In simpler terms, it helps you understand if the relationship between two variables is due to chance or if it's a real relationship.

The test compares the observed frequencies (the data you actually collected) with the expected frequencies (the data you would expect to see if there were no relationship between the variables).

There are two main types of Chi-Square tests:

1. The Chi-Square Goodness of Fit Test: This test is used to compare the observed frequencies of a single categorical variable to its expected frequencies. It tests if the sample distribution fits a claimed population distribution.

2. The Chi-Square Test for Independence: This is the most common type. It's used to determine whether two categorical variables are independent of each other.

The core of the test is the Chi-Square statistic, which is calculated based on the differences between observed and expected counts for each category.

Formula: χ² = Σ [ (Observed - Expected)² / Expected ]

You calculate this value for every category and then sum them up. A large Chi-Square value means there is a large difference between your observed and expected data, suggesting the variables are not independent.

This calculated χ² value is then compared to a critical value from a Chi-Square distribution table (or used to find a p-value) to determine statistical significance.

Like many hypothesis tests, the Chi-Square test gives you a p-value.

The p-value is the probability of observing your data (or something more extreme) if the null hypothesis is true (i.e., if there is no relationship between the variables).

If p-value ≤ α (significance level, usually 0.05): You reject the null hypothesis. This means there is a statistically significant association between the two variables.

If p-value > α: You fail to reject the null hypothesis. This means you do not have enough evidence to conclude that an association exists between the variables.

The Chi-Square test is widely used in business and social sciences.

Market Research: A company can use it to determine if a preference for a new product design is related to the age group of the consumer.

Medicine: Researchers can test if a new drug is more effective than a placebo by comparing the categorical outcomes (e.g., 'improved' vs. 'not improved') for the two groups.

Sociology: An analyst might test if there is a relationship between a person's income bracket and their level of job satisfaction.