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All courses Math I · S-ID.5 53 of 59
Summarize two-category data with two-way tables and interpret joint, marginal, and conditional relative frequencies.

Complete a two-way frequency table from raw counts

Problem
Use these raw counts to complete the two-way frequency table: 30 students: 8 prefer cats and own pets, 7 prefer cats and do not own pets, 10 prefer dogs and own pets, 5 prefer dogs and do not own pets.
A genuine two-way matrix containing only the given information, with requested cells or totals left unresolved. Open full size
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Hint

Use pet preference for one direction and pet ownership for the other, then match each raw count to the cell named by both labels.

A two-way table cell records one combined category, such as cats and own pets or dogs and do not own pets.

Solution walkthrough

01

Read the table categories

\[\text{rows}=\text{prefer}~\text{cats},~\text{prefer}~\text{dogs};~\text{columns}=\text{own}~\text{pets},~\text{no}~\text{pets}\]

Each interior cell must combine exactly one preference row with exactly one pet-ownership column.

02

Place the cat-preference counts

\[\text{prefer}~\text{cats}~\text{and}~\text{own}~\text{pets}=8;~\text{prefer}~\text{cats}~\text{and}~\text{no}~\text{pets}=7\]

The prompt gives 8 students in the cats-and-own-pets combination and 7 in the cats-and-no-pets combination.

03

Place the dog-preference counts

\[\text{prefer}~\text{dogs}~\text{and}~\text{own}~\text{pets}=10;~\text{prefer}~\text{dogs}~\text{and}~\text{no}~\text{pets}=5\]

The remaining two stated groups belong in the dog-preference row under their matching ownership columns.

04

Check the survey total

\[8+7+10+5=30\]

The four disjoint table cells add to the stated 30 surveyed students, so every student is counted once.

05

Report the completed interior table

\[\text{cats}~\text{own}~\text{pets}=8,~\text{cats}~\text{do}~\text{not}~\text{own}~\text{pets}=7,~\text{dogs}~\text{own}~\text{pets}=10,~\text{dogs}~\text{do}~\text{not}~\text{own}~\text{pets}=5\]

Each reported value names both its row and its column, preventing a row-column reversal.

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Another way

  1. Make four labels first—cats/own, cats/no, dogs/own, dogs/no—then copy each raw count to its matching label and verify the total.

!

Common mistake

Do not place 7 under 'own pets' just because it appears second in a sentence. Match both words in the group description to the row and column headers.

Completed two-way matrix with row and column categories, counts or relative frequencies, and relevant totals or denominator scope.
table: cats own pets=8, cats do not own pets=7, dogs own pets=10, dogs do not own pets=5