Conditional Frequencies & Association — I can use conditional relative frequencies to assess association between categorical variables.

People have always wanted to know whether two things are connected. Does studying more lead to better grades? Does a new medicine actually reduce illness? For centuries, scholars relied on intuition and anecdotes, but those approaches were unreliable. The development of two-way frequency tables and conditional relative frequencies gave us a rigorous, numerical way to assess whether categorical variables are truly associated or merely appear that way by coincidence.

The central question this lesson addresses is straightforward but powerful: when you know which group someone belongs to, does that change the likelihood of their response? If yes, the two variables are associated. If no, they are independent. Learning to calculate and compare conditional relative frequencies gives you the tool to answer this question with data instead of guesswork.

Before diving into calculations, you need to understand a few foundational ideas. A categorical variable is one whose values are labels or categories rather than numbers — for example, "favorite sport" or "grade level." When you have two categorical variables measured on the same group of individuals, you can organize the counts in a two-way frequency table (also called a contingency table). Each cell shows how many individuals fall into a particular combination of categories.

The diagram below shows a two-way frequency table for 200 students surveyed about their preferred study method (solo vs. group) and whether they earned an A on a recent exam. The marginal totals appear in the rightmost column and bottom row. Pay close attention to how the joint frequencies in the interior cells add up to the marginal frequencies on the edges.

Notice that the joint frequencies in each row add up to that row's marginal total, and the joint frequencies in each column add up to that column's marginal total. The grand total in the bottom-right corner equals the sum of all joint frequencies and also equals the sum of any complete set of marginal totals. This structure is the foundation for every calculation that follows.

There are three types of relative frequencies you can compute from a two-way table. Understanding the difference is the key to detecting association.

The real power of conditional relative frequencies lies in comparison. Once you calculate conditional relative frequencies for each row (or each column), you line them up side by side. If the percentages are roughly the same across rows, the variables appear to be independent — knowing someone's row category doesn't help you predict their column category. If the percentages differ noticeably, the variables are associated.

A school surveyed 250 students and staff members about which type of phone they prefer: Android or iPhone. The results are organized by age group (Under 18 and 18 or Older). Let's determine whether phone preference is associated with age group.

The conditional relative frequency analysis you learn here is an informal, visual approach. In AP Statistics and college-level courses, you'll encounter formal tools that build directly on the same ideas.

What you're really learning here is the conceptual foundation of conditional probability. When you compute 48 ÷ 160 and interpret it as "the probability of preferring Android given that the person is under 18," you're using the same logic behind P(A | B) = P(A ∩ B) ÷ P(B). Mastering conditional relative frequencies now will make the transition to formal probability and inference much smoother.

Use the following two-way table for Problems 1–5. A school collected data from 300 students about their participation in extracurricular activities and their grade in English class.