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How do you read a two-way frequency table, and what is the difference between joint, marginal, and conditional relative frequencies?

Summarize categorical data in two-way frequency tables and interpret joint, marginal, and conditional relative frequencies, recognizing possible associations (Ohio S-ID.5).

An Ohio Algebra I answer on two-way frequency tables (S-ID.5): reading counts and totals, computing joint, marginal, and conditional relative frequencies, and judging whether two categorical variables are associated.

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  1. What this topic is asking
  2. Reading the table
  3. Joint, marginal, and conditional relative frequencies
  4. Judging association
  5. How Ohio examines this topic
  6. Why the denominator decides the type
  7. Why conditional frequencies, not counts, show association
  8. Try this

What this topic is asking

Ohio standard S-ID.5 asks you to summarize categorical data in a two-way frequency table and to interpret joint, marginal, and conditional relative frequencies, then judge whether the two variables are associated. This is the categorical-data skill in the Statistics category, distinct from the numerical displays, and it shows up as table-reading items.

Reading the table

A two-way table organizes counts by two categories at once.

A quick check: every row should sum to its row total, every column to its column total, and all of those to the grand total.

Joint, marginal, and conditional relative frequencies

A relative frequency is a count expressed as a fraction (or percent). The three types differ only in the denominator.

Judging association

Two categorical variables are associated if knowing one changes the likelihood of the other. You test this by comparing conditional relative frequencies across the categories of one variable.

How Ohio examines this topic

  • Numeric response. Compute a joint, marginal, or conditional relative frequency.
  • Multiple choice and multiple-select. Identify which type a given fraction is, or read a count or total.
  • Tables and drag and drop. Complete a two-way table, or fill in totals.

Why the denominator decides the type

The single most important idea for two-way tables is that joint, marginal, and conditional relative frequencies use the same numerator counts but different denominators, and the denominator is what gives each its meaning. Dividing by the grand total asks "what share of everyone?", that is joint (for an inner cell) or marginal (for a margin total). Dividing by a row or column total asks "what share within this one category?", that is conditional. So 20100\frac{20}{100} and 2040\frac{20}{40} use the same 2020 pet-owning apartment dwellers but answer different questions: the first is the joint rate among everyone, the second is the conditional rate among apartment dwellers. Reading the question to find what it conditions on, and choosing the matching denominator, is the key to every two-way-table item.

Why conditional frequencies, not counts, show association

It is tempting to judge association by comparing raw counts, but counts are misleading when the groups are different sizes. If one group has far more people, it can have more of a trait simply because it is bigger, not because the trait is more common there. Conditional relative frequencies fix this by converting each group's count to a rate within that group, putting them on equal footing. Comparing those rates, the percentage of each category that has the trait, is what reveals whether the variables move together. Equal conditional rates mean the second variable does not depend on the first (no association); clearly different rates mean it does. This is why the standard emphasizes conditional relative frequencies for spotting trends, and why the test phrases association questions in terms of comparing percentages.

Try this

Q1. A table has grand total 200200 and a "yes-yes" cell of 5050. What is that joint relative frequency? [1 point]

  • Cue. 50200=0.25\frac{50}{200} = 0.25.

Q2. Of 8080 people in a column, 6060 answered "yes." What is the conditional relative frequency of yes within that column? [1 point]

  • Cue. 6080=0.75\frac{60}{80} = 0.75.

Exam-style practice questions

Practice questions written in the style of ODEW exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

Ohio Algebra I EOC (style)2 marksNumeric response. In a survey, 3030 of 5050 students walk to school and 1212 of those walkers are in grade 9. What fraction of walkers are in grade 9 (a conditional relative frequency)?
Show worked answer →

The conditional relative frequency is 1230=0.4\dfrac{12}{30} = 0.4, or 40%40\%.

A conditional relative frequency is computed within a single category, here, among walkers only. Of the 3030 walkers, 1212 are in grade 9, so the fraction is 1230=0.4\frac{12}{30} = 0.4. The denominator is the walker total (3030), not the whole survey (5050), because the question conditions on already being a walker. Choosing the right denominator, the row or column total you are conditioning on, is the whole skill.

Ohio Algebra I EOC (style)2 marksMultiple choice. In a two-way table, the total of a single row divided by the grand total gives which kind of relative frequency? (A) marginal (B) joint (C) conditional (D) cumulative
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The correct answer is (A).

A marginal relative frequency uses a row total or column total (found in the margins of the table) over the grand total, it describes one variable ignoring the other. A joint relative frequency is a single inner cell over the grand total (both categories at once). A conditional relative frequency uses a cell over a row or column total. So a row total over the grand total is marginal.

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