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How do you summarize two categorical variables in a two-way frequency table, and compute joint, marginal, and conditional relative frequencies?

Summarize categorical data for two categories in two-way frequency tables, and interpret joint, marginal, and conditional relative frequencies (TN A1.S.ID.C.5).

A TNReady Algebra I answer on two-way frequency tables (TN A1.S.ID.C.5), reading joint and marginal totals, and computing conditional relative frequencies as a fraction of a row or column.

Generated by Claude Opus 4.89 min answer

Reviewed by: AI editorial process; not yet individually human-reviewed

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  1. What this topic is asking
  2. Reading the table
  3. The three relative frequencies
  4. How TNReady examines this topic
  5. Why the denominator is everything
  6. Try this

What this topic is asking

Standard A1.S.ID.C.5 asks you to summarize two categorical variables in a two-way frequency table and to compute and interpret three kinds of relative frequency: joint (a single cell over the grand total), marginal (a row or column total over the grand total), and conditional (a cell over its row or column total). The graded skill is choosing the right denominator.

Reading the table

A two-way table looks like this (students by sport and music):

Plays music No music Total
Plays sport 18 12 30
No sport 8 12 20
Total 26 24 50
  • Cells (18, 12, 8, 12) are joint frequencies, both categories at once.
  • Margins (30, 20 for sport; 26, 24 for music) are marginal totals, one variable.
  • The corner (50) is the grand total.

The three relative frequencies

  • Joint relative frequency: a cell over the grand total, for example 1850=0.36\frac{18}{50} = 0.36 play both a sport and music.
  • Marginal relative frequency: a margin over the grand total, for example 3050=0.6\frac{30}{50} = 0.6 play a sport.
  • Conditional relative frequency: a cell over its row or column total, for example 1830=0.6\frac{18}{30} = 0.6 of athletes play music.

Note this differs from the earlier 1830\frac{18}{30}: the same cell gives a different conditional frequency depending on whether you condition on the sport row or the music column.

How TNReady examines this topic

  • Multiple choice. Choose the correct fraction for a joint, marginal, or conditional frequency, with wrong-denominator distractors.
  • Numeric response. Compute a relative frequency or a total.
  • Drag and drop. Complete a partially filled two-way table using the totals.

A clarifying idea is that the wording picks the denominator: "of all students" means the grand total (joint or marginal), while "of students who play a sport" means a row total (conditional). Reading the condition phrase carefully is the whole skill.

Why the denominator is everything

Almost every error on this topic is a wrong-denominator error, so it is worth understanding why the choice matters. A joint relative frequency answers "what share of everyone," so it divides by the grand total. A conditional relative frequency answers "what share of a subgroup," so it divides by that subgroup's total, a row or a column. The same count of 1818 "sport and music" students becomes 1850\frac{18}{50} (of everyone), 1830\frac{18}{30} (of athletes), or 1826\frac{18}{26} (of musicians) depending on the question. Filling out a table can also reveal association: if the conditional frequency of playing music is much higher among athletes than non-athletes, the two variables appear related. The reliable method is to underline the conditioning phrase ("of those who ...") to fix the denominator before you compute, which turns a confusing item into a simple fraction.

Try this

Q1. Using the table above, what fraction of all students play no sport? [1 point]

  • Cue. Marginal: 2050=0.4\frac{20}{50} = 0.4.

Q2. Of students who play no sport, what fraction play music? [1 point]

  • Cue. Conditional on the no-sport row (2020): 820=0.4\frac{8}{20} = 0.4.

Exam-style practice questions

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

TNReady (style)2 marksMultiple choice. In a survey, of 5050 students, 3030 play a sport and of those 1818 also play music; 88 of the non-athletes play music. What fraction of students who play a sport also play music? (A) 1830\frac{18}{30} (B) 1850\frac{18}{50} (C) 1826\frac{18}{26} (D) 3050\frac{30}{50}
Show worked answer →

The correct answer is (A).

This is a conditional relative frequency: "of those who play a sport." The condition restricts you to the 3030 athletes, and 1818 of them play music, so the fraction is 1830=0.6\frac{18}{30} = 0.6. Distractor (B) divides by the whole 5050 (a joint frequency), and (C) uses the wrong total. The phrase "of students who play a sport" tells you the denominator is the sport row's total.

TNReady (style)2 marksNumeric response. A two-way table shows 4040 adults: 2424 prefer coffee and 1616 prefer tea; of the coffee group, 1010 are under 30. What is the marginal total preferring coffee?
Show worked answer →

The marginal total preferring coffee is 2424.

A marginal frequency is a row or column total, found in the table's margins. The number preferring coffee, regardless of age, is 2424. The "1010 under 30" is a joint frequency (two conditions at once), not the marginal. Marginal totals describe one variable alone; joint frequencies describe the overlap of two.

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