Conditional Relative Frequencies — I can identify possible associations between two categorical variables using conditional relative frequencies (intro).

Long before modern statistics existed, people noticed that certain characteristics seemed to appear together. For instance, a doctor in the 1800s might observe that patients who worked in coal mines were more likely to develop lung disease than patients who worked outdoors. But how could someone move beyond a gut feeling and actually measure whether two categories were truly related? This question drove the development of tools for analyzing categorical data — data that falls into groups or labels rather than numerical measurements.

The fundamental question that conditional relative frequencies answer is deceptively simple: Does knowing which group someone belongs to change the likelihood of a particular outcome? If it does, we say there may be an association between the two variables. If it doesn't, we say the variables appear to be independent. Learning to calculate and interpret conditional relative frequencies gives you a powerful, accessible tool for spotting these patterns.

Before diving into calculations, you need to understand several foundational ideas. A categorical variable is any variable whose values are labels or groups rather than numbers — for example, favorite subject (Math, Science, English) or transportation method (Bus, Car, Walk). A two-way frequency table organizes data about two categorical variables into rows and columns, with counts showing how many data points fall into each combination.

The diagram below shows a two-way frequency table that tracks 200 students by their preferred sport (Soccer or Basketball) and whether they prefer morning or afternoon classes. Notice how the joint frequencies sit inside the table, while the marginal frequencies appear along the right edge and bottom.

This table is the starting point for every conditional relative frequency calculation. The joint frequencies (like 60 soccer-morning students) tell you how many observations share a specific pair of categories. The marginal frequencies (like the 90 total soccer students) tell you the overall size of a single group. To find a conditional relative frequency, you'll divide a joint frequency by the appropriate marginal frequency, which effectively zooms in on one group and asks what that group looks like.

There are two main ways to compute conditional relative frequencies: by row and by column. Each approach conditions on one of the two categorical variables and reveals how the other variable is distributed within each group.

The real power of conditional relative frequencies lies in comparing them across rows or across columns. If the distributions look roughly the same regardless of which row (or column) you examine, the variables are likely independent. If the distributions differ noticeably, there is a possible association. Let's see this in action using the sport vs. class time data from Section 3.

Notice the key comparison in the diagram above. Among soccer students, about two-thirds prefer morning classes. Among basketball students, fewer than half prefer morning classes. This roughly 21-percentage-point gap (66.7% versus 45.5%) suggests that sport preference and class time preference may be associated. If the two variables were independent, the morning percentage would be approximately the same in both rows — close to the overall rate of 110 ÷ 200 = 55%.

A school surveys 300 students about their preferred study method (Group Study or Solo Study) and whether they scored above or below the class median on the last test. The results are shown below:

Conditional relative frequencies provide a foundation for more advanced statistical reasoning. Understanding them prepares you for formal hypothesis testing and deeper work with categorical data. The table below compares what you're learning now with where the topic leads.

The skills you build here — constructing two-way tables, computing conditional percentages, and comparing distributions — are exactly the skills you'll need when you encounter the chi-square test. In that test, you'll compute expected frequencies (what the table would look like if the variables were truly independent) and then measure how far the observed data deviates from those expectations. Everything starts with the conditional relative frequency thinking you're developing right now.

Use the following two-way table for Problems 1–4. A school cafeteria surveyed 250 students about their lunch choice (Pizza or Salad) and their grade level (Freshman or Senior).