Mathématiques

Two-Way Table: Reading Two Categories at Once

A two-way table splits one group by two categories at once. Fill the gaps from the row and column totals, then read a probability straight off the table.

Two-way table

A two-way table sorts one group of people or things by two different categories at once, showing how many fall into each combination.

Aussi appelé: Two-way frequency table, Contingency table
Où les élèves le rencontrent: Grade 6 mathematics, when data is first sorted into categories, and again at GCSE, where two-way tables are the usual setting for probability questions on both Foundation and Higher papers.

La réponse en bref
A two-way table sorts one set of data by two categories at once, such as boys and girls against left- and right-handed. Each inner cell counts the items in both categories, the outer edges hold the row and column totals, and any missing cell can be recovered from those totals.

Un exemple

Left-handed girls = 32 − 25 = 7

Sixty students are recorded. The table says 32 are girls, of whom 25 are right-handed, so 7 girls are left-handed. It also says 28 are boys, of whom 4 are left-handed, so 24 boys are right-handed. The left-handed column now totals 11, the right-handed column 49, and 11 + 49 returns the 60 the table started with. Nothing here needs a formula; it needs every line to add up.

The middle counts, the edges total

Every cell inside a two-way table answers one question: how many things are in this row category and this column category together. A cell reading 25 in the girls row and the right-handed column means 25 people who are both. Reading a cell without reading both of its labels is where most mistakes start.

The outer row and column are different in kind. They are not another category, they are the totals, and the number in the far corner is the total of everything. That corner number appears twice over if you add the row totals and again if you add the column totals, which is the built-in check on the whole table.

Completing a table you have been given half of

The method is arithmetic with a route. Every row adds to its row total, every column adds to its column total, and the row totals and column totals both add to the grand total. Find whichever line has exactly one number missing, fill it, and that usually opens up the next line.

Work in this order and a half-empty table falls out in a couple of minutes:

Look for a row or column with one gap and no other unknowns, and subtract
Fill it in, then check whether that completes a line running the other way
Keep going until only the totals are left, and calculate those last
Check twice at the end: the row totals must add to the grand total, and so must the column totals

The probability question, and which total goes underneath

Once the table is complete, a probability is one number from the table over another number from the table. The whole difficulty is deciding what belongs on the bottom. Picking a student at random from all sixty gives P(left-handed) = 11/60, because the group being picked from is everybody.

Change the wording to "a left-handed student is chosen" and the group being picked from shrinks. The probability that student is a girl is 7/11 — the left-handed column total, not 60. Phrases like "given that", "of the left-handed students" and "one of the girls" are all instructions about the denominator, and they are the difference between one mark and three.

Questions fréquentes

How do you complete a two-way table?

By subtraction, starting wherever a row or column has only one number missing. Fill that gap, then look across the table to see which line the new number completes. Leave the totals until last so they act as a check rather than as another guess, and confirm at the end that the rows and the columns both add to the same grand total.

How do you work out a probability from a two-way table?

Put the number of items you want on top and the size of the group being chosen from underneath. If the question picks from everyone, the denominator is the grand total. If it picks from one row or one column, the denominator is that line's total instead. Reading the question for which group is being sampled matters more than the arithmetic.

What is the difference between a two-way table and a frequency table?

A frequency table records one thing about each item, such as how many students got each grade. A two-way table records two things at once and shows how they combine. That extra dimension is what lets it answer questions a frequency table cannot, such as whether girls and boys got different grades.

Why do my rows and columns not add up to the same total?

Almost always because something has been counted twice or not at all. A two-way table only works if each item falls into exactly one row category and exactly one column category, so overlapping options such as "walks" and "cycles" for a student who does both will break it. Check the categories before checking the arithmetic.

Dernière mise à jour 21 août 2026