Two-way tables
A two-way table is a way to organise data about two specific variables. This example shows a two-way table where the variables are gender and favourite sport.
The table holds a lot of information. For example, we can see that:
- the total number of people sampled is 60
- 20 people like rugby
- seven girls like football
We can also explore the probabilities related to each group on the table.
Example one
What is the probability of randomly selecting someone who likes rugby?
There are 20 people who like rugby, and there are 60 people in total, so the probability is \(\frac{20}{60}\) = \(\frac{1}{3}\).
Example two
If a girl is selected at random, what is the probability that she likes a sport other than football or rugby?
There are 28 females, 13 of which prefer sports other than football or rugby, so the probability is \(\frac{13}{28}\).
Completing a two-way table
Often you will be asked to complete a partially filled two-way table. We can do this by taking advantage of the fact that each row and column has a total associated with it.
The table below shows the income of people compared to their age – but there are numbers missing. We can fill in any gap provided it is the only missing number in the row or column.
The sections shaded yellow or blue contain missing numbers. The yellow cells can be worked out immediately but it is impossible to fill in the blue cell without first completing the others around it, this is because we do not yet have enough information. If we want to work out the blue cell we must work out either the yellow cell to its left, or the yellow cell below it first.
To work out the missing cell in the 'Less than £20k' column we simply add them together to find the missing total:
24 + 7 + 3 = 34
To work out the missing cell in the '£20k-40k' column we must subtract the 12 and the 9 from 33:
33 – 12 – 9 = 12
To work out the missing cell in the '£40-60k' column we have to look across the row instead. Using the 40+ row we can see that we must subtract 3, 9 and 5 from 31:
31 – 3 – 9 – 5 = 14
We now have two ways to work out the missing number in the blue cell. We can either subtract 7, 12 and 2 from 33:
33 – 2 – 7 – 12 = 12
or we can subtract 2 and 14 from 28:
28 – 2 – 14 = 12
The completed table now looks like this:
