Summarize categorical data in a two-way frequency table and interpret joint, marginal, and conditional relative frequencies (LA A1: S-ID.B.5).

What this topic is asking

Standard A1: S-ID.B.5 asks you to summarize categorical data in a two-way frequency table and interpret joint, marginal, and conditional relative frequencies. On LEAP 2025 these are Type I and Type II items in the Additional and Supporting Content category, often reading a table and choosing the correct denominator.

Reading the table

A two-way table has rows for one variable's categories and columns for the other's. Each cell is a count for one combination; the margins hold the row and column totals; the bottom-right corner is the grand total.

The three relative frequencies

The denominator is what distinguishes them: grand total for joint and marginal, group total for conditional.

Working an example

Note this is different from "of seniors, the fraction who prefer pizza" ($\frac{24}{60} = 0.40$): the condition sets the denominator.

How LEAP examines this topic

• Equation response. Compute a joint, marginal, or conditional relative frequency.
• Type II reasoning. Interpret a frequency, or decide whether two categories appear associated.
• Drag and drop. Complete a two-way table's missing cells or totals.

A clarifying idea: comparing conditional frequencies across groups reveals association. If juniors prefer pizza 75 percent of the time but seniors only 40 percent, the preference differs by class, evidence the two variables are related.

Why the denominator decides the meaning

The single most important idea in two-way tables is that the denominator defines the question, which is exactly what S-ID.B.5 tests. A relative frequency is always a part over a whole, but the table offers three different "wholes," and choosing among them changes what the number means. Dividing by the grand total asks "what fraction of everyone?", a joint or marginal view that describes the population as a whole. Dividing by a row or column total asks "what fraction of this particular group?", a conditional view that zooms into one category. These can be very different numbers from the same cell: $\frac{30}{100} = 0.30$ of all students are pizza-preferring juniors, but $\frac{30}{40} = 0.75$ of juniors prefer pizza. Confusing them is the classic error, and it matters because conditional frequencies are how you detect association between the variables: you compare the conditional rate in one group to the conditional rate in another, and a real difference suggests the variables are related. Marginal frequencies, by contrast, ignore the second variable entirely and just describe one variable's distribution. Reading a two-way table well is therefore mostly about pausing to ask "fraction of what?" before dividing, and that habit is the skill the standard rewards.

Try this

Q1. From the table, what is the marginal relative frequency of seniors? [1 point]

• Cue. $\frac{60}{100} = 0.60$.

Q2. From the table, of the seniors, what fraction prefer tacos? [2 points]

• Cue. $\frac{36}{60} = 0.60$ (column-restricted to seniors' total).