Why Does the Chi-Squared Test Work?

The chi-squared test (χ2 test) is a fundamental statistical technique used to determine if there is a significant association between two or more categorical variables. This article explains the core principles and steps behind the chi-squared test, aimed at anyone interested in understanding this powerful analytical tool.

1. Comparison of Observed and Expected Frequencies

Observed and expected frequencies are the two crucial components of the chi-squared test. The observed frequencies represent the actual counts from the data, while the expected frequencies are the counts we would anticipate if there were no correlation between the variables. The difference between these two sets of frequencies is then quantified using the chi-squared statistic (χ2).

The formula for the chi-squared statistic is given by:

χ2 Σ [(O_i - E_i)2 / E_i]

Where:

O_i: Observed frequency for the ith category E_i: Expected frequency for the ith category

This formula helps to identify whether the observed data significantly deviates from the expected data, indicating a potential association between the categorical variables.

2. Distribution of the Chi-Squared Statistic

Under the null hypothesis, which suggests no association between the variables, the chi-squared statistic follows a chi-squared distribution. The shape of this distribution is determined by the degrees of freedom, which are calculated as the number of categories minus 1 in each variable minus 1.

3. Null and Alternative Hypotheses

The chi-squared test involves two hypotheses:

Null Hypothesis (H?): Assumes no association between the categorical variables. Alternative Hypothesis (H?): Assumes there is an association between the categorical variables.

4. Significance Level and P-Value

After calculating the chi-squared statistic, it is compared to a critical value from the chi-squared distribution at the chosen significance level, typically 0.05, to determine if the null hypothesis should be rejected. Alternatively, a p-value can be calculated, which indicates the probability of observing the calculated chi-squared statistic or a more extreme one, assuming the null hypothesis is true. A p-value less than the significance level (alpha) leads to the rejection of the null hypothesis.

5. Assumptions

To ensure the validity of the chi-squared test, several assumptions must be met:

The data must be in the form of counts or frequencies. Observations must be independent. The expected frequency in each category should be sufficiently large, generally at least 5, to ensure the chi-squared distribution remains accurate.

Summary

The chi-squared test quantifies the difference between observed and expected frequencies using the properties of the chi-squared distribution. This allows it to assess whether these differences are statistically significant, making it a robust tool for analyzing categorical data across various disciplines, including biology, social sciences, and market research.