χ²-tests — common mistakes

A common exam question involves a distribution where one or more expected frequencies are less than 5. You must state that you are combining (or 'pooling') adjacent categories and then recalculate the degrees of freedom based on the new, smaller number of categories. Forgetting to do this is a frequent source of lost marks.

Always show your table of expected frequencies in your working. Marks are often awarded for this. When writing your conclusion, make sure it is stated in the full context of the problem, referring to the variables involved (e.g., 'association between age and film preference') rather than just 'reject H₀'.

A goodness of fit test compares one set of observed categorical data (from one variable) against a theoretical distribution. A test for independence uses a two-way contingency table to see if there is a relationship between two different categorical variables.

You must combine that category (or cell) with an adjacent one. For a goodness of fit test, you might combine '4 or more defects'. For a contingency table, you might combine two rows or two columns. After combining, you must use the new number of categories/rows/columns to calculate the degrees of freedom. You should always state which categories you have combined.

The degrees of freedom reflect the number of values that are 'free to vary'. In a goodness of fit test, once you've filled (k-1) categories and you know the total, the last category is fixed. So ν starts at k-1. In a contingency table with 'r' rows and 'c' columns, once you've filled in (r-1) rows and (c-1) columns, the remaining cells are all fixed by the row and column totals. This gives (r-1)(c-1) free choices.