what is a chi-square test?

Sourav PanSeptember 10, 2024

Answered

A Chi-Square test is a statistical method used to evaluate the relationship between categorical variables or to assess how well observed data fit a theoretical distribution. It is widely used in hypothesis testing to determine if there are significant differences between expected and observed frequencies in categorical data. Here’s a detailed explanation of what a Chi-Square test involves:

Purpose

1. Test of Independence:
   • Objective: To determine if there is a significant association between two categorical variables.
   • Example: Assessing whether gender is related to voting preference (e.g., whether male and female voters show different preferences for candidates).
2. Goodness of Fit Test:
   • Objective: To assess how well observed data match an expected distribution.
   • Example: Testing whether a die is fair by comparing the observed frequencies of each face to the expected frequencies if each face had an equal chance of appearing.
3. Test of Homogeneity:
   • Objective: To compare the distribution of a categorical variable across different populations or groups to see if they are similar.
   • Example: Comparing the distribution of preference for a product across different regions to see if it is consistent.

How It Works

1. Calculate Expected Frequencies:
   • For Independence and Homogeneity Tests: Use the marginal totals of the contingency table to compute the expected frequency for each cell in the table.
   • For Goodness of Fit Test: Use the theoretical distribution to calculate the expected frequencies.
2. Compute the Chi-Square Statistic:
   • Formula: \chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}}
   • 1. • where OiO_iOi​ represents the observed frequency in each category, and EiE_iEi​ represents the expected frequency for that category.
        • Procedure: Sum the squared differences between observed and expected frequencies, divided by the expected frequencies.
     2. Determine Significance:
        • Compare to Critical Value: Compare the Chi-Square statistic to a critical value from the Chi-Square distribution table, based on the desired level of significance (alpha) and degrees of freedom.
        • Calculate p-Value: Alternatively, compute the p-value to assess the significance of the Chi-Square statistic.

     Assumptions

     1. Independence: The observations should be independent of each other.
     2. Sample Size: Typically, expected frequencies in each cell should be 5 or more to ensure the validity of the Chi-Square approximation.

     Applications

     • Social Sciences: Analyzing survey data to identify associations between demographic factors and opinions.
     • Biology: Testing genetic data to see if observed allele frequencies fit expected Mendelian ratios.
     • Marketing: Evaluating consumer preferences across different market segments.