Summarizing Categorical Data
Categorical data places each individual into a group instead of measuring a number — plays a sport or not, owns a bike or not, grade level. When you track two categories at once, the standard tool is a two-way frequency table: rows for one category, columns for the other, and a count in every cell.
Almost every two-way table question comes down to knowing which number to grab: an inner cell, a row or column total, or one of those divided by something. Learn the four frequency names — joint, marginal, relative, and conditional — and the questions read themselves.
Reading a two-way table
Each inner cell is a joint frequency: the count of individuals in both categories at once, like 7th graders who do not own a bike. Find it by following the row and the column until they meet.
The row and column totals are marginal frequencies — they live in the margins of the table and count one category alone, ignoring the other. The bottom-right corner is the grand total, the number of individuals in the whole table.
Relative and conditional frequencies
A relative frequency divides a count by the grand total, turning it into a fraction or decimal of the whole group. If of students play both a sport and an instrument, the relative frequency is .
A conditional frequency restricts the question to one row or column first. "Of the students who play a sport, what fraction also play an instrument?" divides by the sport row's total, not the grand total: . The phrase "of the..." tells you which total goes in the denominator.
The two-way table below holds those numbers: the inner cell is the joint frequency (sport and instrument), the row total is the marginal frequency for playing a sport, and the corner is the grand total.
Worked examples
Example 1: read a joint frequency
A table of 60 students shows bike ownership by grade. The 7th-grade row reads: Bike , No Bike , Total . How many 7th graders own a bike?
Answer: students
Example 2: find a relative frequency
In a survey of students, play both a sport and an instrument. What is the relative frequency of playing both?
Answer: , or
Example 3: find a conditional frequency
Of the students who play a sport, also play an instrument. Of the students who play a sport, what fraction also play an instrument?
Answer: , or
Try one yourself
Common questions
What is the difference between a joint and a marginal frequency?
A joint frequency is an inner cell — it counts individuals in two categories at once. A marginal frequency is a row or column total in the table's margin — it counts one category and ignores the other.
How do I know whether to divide by the grand total or a row total?
Read the question's opening phrase. A plain relative frequency divides by the grand total. If the question starts with "of the students who...", it is conditional — divide by that group's row or column total instead.
How can I check that a two-way table is filled in correctly?
Every row's cells should add to its row total, every column's cells should add to its column total, and both the row totals and the column totals should add to the same grand total. If any of those fail, a cell is wrong.
Why use a two-way table instead of two separate lists?
Because the interesting question is usually about the combination — whether sport players are more likely to play an instrument, say. Two separate one-category counts cannot show a relationship; the joint cells can.
Want the video version?
Allday Everyday Math has video lessons, practice, and an AI tutor for every topic, Pre-Algebra through Algebra 2.