Chi-square tests are really interesting and useful tools in statistics! They help us figure out if there is a meaningful connection between different categories or if what we see in our data matches what we expect. There are two main types of chi-square tests: the Goodness of Fit test and the Independence test.

Goodness of Fit Test

• Purpose: This test checks if what we see (the observed frequencies) matches what we expect to see (the expected frequencies).
• Example: Think about rolling a six-sided die. You would use this test to find out if each number shows up about 1 out of 6 times like we would expect.

Independence Test

• Purpose: This test looks at whether two categories are related or not.
• Example: An example is looking into whether there is a connection between a person’s gender and their choice of a favorite product.

What’s great about the chi-square statistic is that it's easy to understand. You can calculate it with this simple formula:

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

In this formula, $O_i$ means the frequencies we actually observe, and $E_i$ means the frequencies we expect. A higher chi-square value usually means there is a stronger connection between the categories or that what we observe doesn’t match our expectations very well.

In simple terms, chi-square tests help us make smart guesses about our data, and that’s what inferential statistics is all about!