Chi-square tests are important tools for looking at relationships in data that can be grouped into categories. They help us see if the numbers we observe are different from what we would expect.

1. Types of Tests:

   • Chi-square Goodness-of-Fit: This test checks if the way a single category is spread out matches what we thought it would be.
   • Chi-square Test of Independence: This test looks at whether two categories are related or if they stand alone.

2. Key Statistics:

   • Chi-square statistic: This is calculated with the formula:

     $X^2 = \sum \frac{(O_i - E_i)^2}{E_i}$

     Here, (O_i) stands for the observed frequency (the actual counts we see) and (E_i) is the expected frequency (the counts we thought we would see).

   • Degrees of Freedom: This helps determine the number of categories we can use. The formula is:

     $df = (r - 1)(c - 1)$

     where (r) is the number of rows and (c) is the number of columns in your data.

3. Significance Level:

   • A p-value less than 0.05 usually means there’s a significant relationship between the variables.

   • We can also measure effect sizes (which show how strong the relationship is) using Cramér's V:

     $V = \sqrt{\frac{X^2}{n \cdot \min(k-1, r-1)}}$

     Here, (n) is the total number of responses, and (k) is the number of categories.

All these parts work together to help us understand the connections in categorical data, especially in research about psychology.