Math

Two-Way Tables

2 min readEasy5-question drill

Two-way tables show up on almost every test, and they're basically free points once you know how to read them. The trick is figuring out exactly which number goes on top of the fraction and which goes on the bottom.

Where this lesson fits

Two-Way Tables

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A two-way table is just a grid that sorts data by two categories at once. The rows are one category, the columns are another, and each cell counts how many things fit both labels.

Here's a survey of 200 people about coffee vs. tea, split by age:

Coffee vs. Tea by age (n = 200)

	Coffee	Tea	Total
Under 30	60	40	100
30 or older	50	50	100
Total	110	90	200

Inner cells are counts in both categories; the last row/column are marginal totals; 200 is the grand total.

Let's name the parts:

The rows here are age groups (Under 30, 30 or older).
The columns are drink preference (Coffee, Tea).
Each inner cell is a count — e.g. 60 people are both under 30 and coffee drinkers.
The Total row and column are called marginal totals (the margins of the table). The bottom-right number (200) is the grand total — everyone.

Every two-way table question is really a fraction question. The whole game is choosing the right numerator (top) and denominator (bottom).

Step 1 — Find the count you care about (numerator). This is the specific group the question asks about, e.g. "people under 30 who prefer tea" → 40.

Step 2 — Find the right total (denominator). This is where students lose points. Read the question carefully:

"Of all people surveyed..." → denominator is the grand total (200).
"Of people under 30..." → denominator is that row total (100).
"Of coffee drinkers..." → denominator is that column total (110).

Step 3 — Divide, then convert to a percent (multiply by 100) or leave as a fraction if asked.

probability or fraction = (count in the cell) / (the correct total)

The phrase right after "of" usually tells you the denominator. "What fraction of the people under 30 chose tea?" → the words "under 30" set the denominator to 100, and "tea" picks the cell, 40. So 40/100 = 0.40 = 40%.

That's the whole topic. The numbers are easy; the only difficulty is matching the question's wording to the right total. Underline the group named after "of" before you compute anything.

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Quick check

A bag contains 3 red marbles, 5 blue marbles, and 2 green marbles. If one marble is selected at random, what is the probability that it is blue?

1/2

1/5

3/10

2/5

Worked examples

Example 1

A bag contains 3 red marbles, 5 blue marbles, and 2 green marbles. If one marble is selected at random, what is the probability that it is blue?

Example 2

A two-way table shows the results of a survey of 200 people about their preference for coffee or tea, categorized by age group.

	Coffee	Tea	Total
Under 30	60	40	100
30 or older	50	50	100
Total	110	90	200

Of the people surveyed who prefer tea, what fraction are 30 or older?

Example 3

A two-way table records 80 students by grade level and whether they ride the bus.

	Bus	No Bus	Total
Freshman	18	12	30
Senior	14	36	50
Total	32	48	80

What percentage of freshmen ride the bus?

Common pitfalls

Using the grand total when a row/column is meant

If the question says "of the seniors" or "of coffee drinkers," the denominator is that row or column total — NOT 200 or whatever the grand total is. Underline the group named after the word "of."

Grabbing a marginal total instead of a cell

When a question asks for people in two categories (e.g. "under 30 AND tea"), the answer is an inner cell, not a row or column total. Find where the row and column intersect.

Forgetting to convert fraction to percent

If the answer choices are percents, you must multiply your decimal by 100. 0.55 becomes 55% — leaving it as 0.55 or as 110/200 may not match a choice.

Mixing up which total a partial count belongs to

Double-check that the cell you picked actually lies in the row/column of your denominator. A senior bus-rider (14) does not belong over the freshman total (30).

Key takeaways

Every two-way table question is a fraction: (the count you want) / (the correct total).
The category named right after the word 'of' sets the denominator (grand total, a row total, or a column total).
Inner cells count things in BOTH categories; margin totals count just one category.
Convert to a percent by multiplying the decimal by 100 when the choices are percents.
Underline the group and the condition before computing to avoid using the wrong total.

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Try it yourself

5 practice questions on Two-Way Tables, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

Two-Way Tables

You are here

→Probability and Conditional Probability →Percentages →One-Variable Data: Distributions and Measures of Center and Spread