Two-Way Tables

A two-way table sorts one group of people or things in two ways at once. One way runs down the side, in the rows. The other runs across the top, in the columns.

Each cell counts the people or things that fit both its row and its column. In a table of ice creams, the cell in the Tub row and the Mint column counts the ice creams that are mint and in a tub.

The Total column adds along each row. The Total row adds down each column. The number in the bottom right corner is the total of everyone in the table.

In the exam you are usually asked to complete a two-way table using information written in words, for 3 marks. A second part may ask for a probability from the table, for 1 or 2 marks. Some longer problems don't mention a table at all, but drawing one is the clearest way in.

Every row adds up to its row total, and every column adds up to its column total. So a two-way table can always be checked in two directions.

To find a missing number, choose a row or column with only one gap. Take the numbers you know away from its total:

\text{missing number} = \text{total} - \text{the numbers you know}

Only add or subtract numbers that are in the same row or the same column. Adding numbers from different rows and columns gives a number that means nothing.

• Look at which labels go down the side and which go across the top.
• Put each given number in its cell. A fact such as “12 of the 28 students who chose kayaking are in Year 10” gives two numbers: a total of 28 and a cell of 12.
• Tick off each fact in the question as you use it.
• Find a row or column with only one gap, and fill it by subtraction. Repeat until the table is full.
• Check that every row and every column adds up, and that the total in the bottom right corner is filled in.

When one person is chosen at random from a group, each person in that group is equally likely to be picked. Then:

\text{probability} = \frac{\text{number who fit the description}}{\text{number in the group you choose from}}

Read the question carefully to see who can be chosen. “One of these people” means everyone, so divide by the total in the bottom right corner. “One of the women” means only the women, so divide by the women’s total.

Give a probability as a fraction, a decimal or a percentage, never as a ratio such as 1 : 4

To find what percentage of a group fit a description, divide by that group’s total, then multiply by 100. If 18 of the 24 Year 10 students walk to school: \frac{18}{24} \times 100 = 75\%

To compare groups of different sizes, compare percentages, not counts. If 21 of the 35 Year 11 students walk, that is \frac{21}{35} \times 100 = 60\%. More Year 11 students walk, but a smaller percentage of them do.

Some questions give facts as percentages or fractions of a group. Work out each count first, then put it in the table. Check which group each percentage is of: 60\% of the boys is a percentage of the boys’ total, not of everyone.

Some longer questions sort a group in two ways but show no table. Draw your own two-way table. Examiners report that students who did this did better than those who wrote numbers all over the page.

Label the rows and columns, fill in what you are told, then work from a row or column with one gap. You may be told that two cells are equal. If they are the only two gaps in a row or column, work out what they add up to, then halve it.

• Putting a number in the wrong cell. Find its row and its column before you write it in, and tick off each fact as you use it.
• Not checking the totals. Check that every row and every column adds up; if one doesn’t, find the slip.
• Leaving the bottom right corner blank. Fill in the total there too.
• Adding numbers from different rows and columns. Only add or subtract numbers in the same row or column.
• Dividing by the wrong total. For a probability, divide by the number in the group that can be chosen.
• Writing a probability as a ratio. Give it as a fraction, a decimal or a percentage.