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Two Way Tables
Here we will learn about two way tables, including what a two way table is, how to construct them, interpret them, and how to calculate the probability of an event using a two way table.
There are also two way tables worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.
What are two way tables?
Two way tables are a type of frequency table used for organising data. They are also known as contingency tables.
Two way tables are mostly used for categorical data, though they can be used for numerical data too. Categorical data is data where the items are words rather than numbers.
For example, colours, sports, and car manufacturers.
To construct a two way table, we need two categorical variables. One variable is featured as the top row within the two way table, and the other variable features on the first column of the table.
For example,
This two way table shows a data set about what students eat for lunch. One categorical variable is the gender of the child, the other categorical variable is what type of lunch they have.
There are row and column totals. So we can see that there are katex is not defined boys and that katex is not defined students have packed lunch. The total number of students in the table is katex is not defined
What are two way tables?
How to construct two way tables
In order to construct two way tables:
- Fill in the known values into the two way table.
- Calculate missing values.
- Calculate the row and column totals.
- Check the final total.
Explain how to construct two way tables
Two way tables worksheet
Get your free two way tables worksheet of 20+ questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREE Two way tables worksheet
Get your free two way tables worksheet of 20+ questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREERelated lessons on representing data
Two way tables is part of our series of lessons to support revision on representing data. You may find it helpful to start with the main representing data lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:
Two way tables examples
Example 1: missing row and column totals
Complete this two way table,
- Fill in the known values into the two way table.
Here, we know each of the individual values for each category so we can move on to Step 2.
2Calculate missing values.
Here, we do not have any missing values and so we can move on to Step 3.
3Calculate the row and column totals.
The total number of boys katex is not defined.
The total number of girls katex is not defined.
The total number of students who like rugby katex is not defined.
The total number of students who do not like rugby katex is not defined.
Filling in these values into the table, we have
4Check the final total.
The total number of boys and girls katex is not defined.
The total number of students who do (not) like rugby katex is not defined.
Filling in the grand total of katex is not defined, we have the final solution.
Example 2: three categories with missing totals
Complete this two way table,
Fill in the known values into the two way table.
Here, we know each of the individual values for each category so we can move on to Step 2.
Calculate missing values.
Here, we do not have any missing values and so we can move on to Step 3.
Calculate the row and column totals.
The total number of males katex is not defined.
The total number of females katex is not defined.
The total number of people who drink tea katex is not defined.
The total number of people who drink coffee katex is not defined.
The total number of people who drink juice katex is not defined.
Filling in these values into the table, we have
Check the final total.
The total number of males and females katex is not defined.
The total number of people who drink tea, coffee, and juice katex is not defined.
Filling in the grand total of katex is not defined, we have the final solution.
Example 3: missing values throughout the table
Complete this two way table,
Fill in the known values into the two way table.
Here, we know some of the individual values for each category so we can move on to Step 2.
Calculate missing values.
We have quite a few missing values in the two way table, so we need to find these values. We need to look for a row or column where we know all but one of the missing values.
Here, we know the number of females who can swim, and the total number of people who can swim. This means we can calculate the number of males by subtracting the number of females from the row total, katex is not defined
Now we can calculate the number of males that cannot swim as we know the number of males that can swim, and the total number of males, katex is not defined.
Now we can calculate the number of females who cannot swim, katex is not defined.
Calculate the row and column totals.
The total number of females = katex is not defined.
Filling in this value, we have
Check the final total.
The total number of males and females katex is not defined.
The total number of people can (not) swim katex is not defined.
Filling in the grand total of katex is not defined, we have the final solution.
Example 4: constructing a two way table from a worded problem
A school is researching the handedness of katex is not defined students in Year katex is not defined students in Class A are right-handed. katex is not defined students in Class B are left-handed. katex is not defined students in Class C are left-handed. There are katex is not defined students who are right-handed, and katex is not defined students in Class B. Construct a two way table to represent this data.
Fill in the known values into the two way table.
Here, we know that we have katex is not defined classes (A, B, C) and we are comparing left and right handedness. Drawing the two way table, we have
Completing the information given from the question, fill in the known values into the correct section in the table.
Calculate missing values.
We have quite a few missing values in the two way table, so we need to find these values. We need to look for a row or column where we know all but one of the missing values.
Here, we know that out of the katex is not defined students in Class B, katex is not defined are left-handed, and so we can find the number of right-handed students in Class B, katex is not defined.
We now know all but one value in the right-handed row, katex is not defined.
For the number of left-handed students in Class A, we need to know the row total, katex is not defined.
The number of left-handed students in Class A katex is not defined.
Calculate the row and column totals.
The total number of students in Class A katex is not defined.
The total number of students in Class C katex is not defined.
Filling in these values, we have
Check the final total.
The total number of students in the three classes katex is not defined.
The total number of left-/right-handed people katex is not defined.
Example 5: using a two way table to work out a probability
katex is not defined students chose to study History or Geography.
katex is not defined of the students chose History.
katex is not defined of the boys chose Geography.
katex is not defined of the girls chose History.
A student is chosen at random. By constructing a two way table, work out the probability that the student is a boy who studies history.
Fill in the known values into the two way table.
As katex is not defined students chose to study History or Geography, this is our overall total.
katex is not defined students chose History.
katex is not defined boys chose Geography.
katex is not defined girls chose History.
This completes the table with all the information we know so far.
Calculate missing values.
Inspecting the table, we need to find the row or column where we only have one unknown value. This is the column for History, where we need to calculate the number of boys who chose History, katex is not defined.
To calculate the number of girls who chose Geography, we need to know the total number of girls, katex is not defined.
The number of girls who chose Geography is therefore katex is not defined.
Calculate the row and column totals.
The total number of boys katex is not defined.
The total number of girls katex is not defined.
Check the final total.
The total number of boys and girls katex is not defined.
The total number of students who chose History or Geography katex is not defined.
Now that we have the two way table, we need to use this to calculate the probability of selecting a student who is a boy who studies History.
Looking at the two way table, there are katex is not defined boys who study History out of a total of katex is not defined students, so katex is not defined
Note: Unless the question states, leave the answer to a probability as a fraction without simplifying it – it may be useful for the next question.
Example 6: conditional probability
People can sit in a seat in the stalls or the circle or the balcony of a theatre. There are katex is not defined seats altogether.
There are katex is not defined children who have seats in the theatre.
There are katex is not defined adults in the stalls and katex is not defined adults in the circle.
Altogether there are katex is not defined people in the stalls and katex is not defined people in the balcony.
A person is chosen at random. Work out the probability that the person is a child, given that they are sitting on the balcony.
Fill in the known values into the two way table.
As there are katex is not defined seats altogether, the overall total is katex is not defined
As there are katex is not defined children, the total number of children is katex is not defined
As there are katex is not defined adults in the stalls, and katex is not defined adults in the circle, we can fill in these two values in the adult row.
As there are katex is not defined people in the stalls and katex is not defined people in the balcony, we can also fill in these values.
Calculate missing values.
The row or column that has one missing value is the column for the stalls where we do not know the number of children, katex is not defined.
To calculate the number of children in the circle, we need to know the total number of people in the circle, katex is not defined.
The number of children in the circle is, katex is not defined.
The total number of children in the balcony is katex is not defined.
The number of adults in the balcony is katex is not defined.
Calculate the row and column totals.
The only total we need to calculate to complete the two way table is the number of adults,
katex is not defined.
Check the final total.
The total number of people katex is not defined
The total number of people in the stalls, circle and balcony katex is not defined
Now we can calculate the probability of selecting a person at random that is a child, given that they are in the balcony seats.
There are katex is not defined people in the balcony seats.
katex is not defined of these people are children.
So katex is not defined.
Common misconceptions
- Whole numbers
The numbers in the two way table are whole numbers, not decimals or fractions. This is because the numbers are frequencies where data has been counted.
- Modelling in maths
The situations in maths questions have been simplified to make the situation simpler to study. The world is a more complex place to live. So individuals need to go under one category and one only.
For example, a school asks students to choose football or rugby. In maths we keep these separate. But in real-life the school may accommodate students who want to choose both or neither. This is where we would use a Venn diagram instead.
Step-by-step guide: Venn diagram
- Frequencies
Each individual is only counted once in the main body of the table. Each of the katex is not defined individuals are only in one of the four cells of the main section of the table.
- Overall total
Be careful not to double the overall total (or grand total).
For example, the ‘grand total’ of this table is katex is not defined. Each individual item only counts once towards the ‘grand total’. But it can be worked out using the row totals or the column totals.
katex is not defined
Practice two way tables questions
1. Complete the two way table.
The column totals are katex is not defined and katex is not defined
The row totals are katex is not defined and katex is not defined
The overall total will be katex is not defined
2. Complete the two way table.
The column totals are katex is not defined and katex is not defined.
The row totals are katex is not defined and katex is not defined.
The overall total will be katex is not defined
3. Complete the two way table.
Find the missing values in the table by subtracting or adding.
Total number of children katex is not defined.
Adult and does not drink tea katex is not defined.
Child and drinks tea katex is not defined.
Child and does not drink tea katex is not defined.
Total number of people who do not drink tea katex is not defined.
4. katex is not defined sportsmen and women were asked about what sport they play. Draw a two way table showing the results below.
- katex is not defined people play football
- katex is not defined people are women
- katex is not defined people play hockey
- katex is not defined men play hockey
- katex is not defined women play rugby
Find the missing values in the table by subtracting or adding.
The information from the question gives us the following incomplete two way table.
The number of females who play hockey katex is not defined.
The number of females who play football katex is not defined.
The number of males who play football katex is not defined.
The total number of rugby players katex is not defined.
The number of males who play rugby katex is not defined.
The total number of males katex is not defined.
5. katex is not defined people were asked if they ate meat or were vegetarian/vegan.
katex is not defined women were asked and 10 of them did not eat meat.
katex is not defined men ate meat.
Calculate the probability of selecting a male at random that does not eat meat.
The original two way table is
Filling in the other values into the two way table, we have
The number of males who do not eat meat is katex is not defined
The total number of people is katex is not defined
katex is not defined
6. katex is not defined students are asked to choose French, German or Spanish.
Altogether katex is not defined students chose German and katex is not defined chose Spanish.
katex is not defined boys chose French.
katex is not defined girls chose German.
There are katex is not defined boys altogether.
Complete the table and use it to work out the probability that a girl selected studies Spanish.
Calculating the values in the two way table, we have
The total number of girls katex is not defined.
The number of girls who study Spanish katex is not defined.
katex is not defined.
Two way tables GCSE questions
1. Complete the two way table:
(3 marks)
katex is not defined
(3)
katex is not defined correct
(2)
katex is not defined correct
(1)
2. katex is not defined students each choose to study Drama or Music.
katex is not defined of the students choose Drama.
katex is not defined boys choose Music.
katex is not defined girls choose Drama.
(a) Use this information to complete the two way table.
(b) A student is picked at random. Work out the probability that the student is a boy who chooses drama.
(c) A Music student is picked at random. Work out the probability that the student is a girl.
(8 marks)
(a)
(1)
katex is not defined
(3)
katex is not defined correct
(2)
katex is not defined correct
(1)
(b)
katex is not defined or katex is not defined
(1)
katex is not defined(1)
(c)
katex is not defined or katex is not defined
(1)
katex is not defined(1)
3. katex is not defined students are asked to choose one revision session to attend.
The revision classes are – Biology or Chemistry or Physics.
There are katex is not defined male students.
katex is not defined students choose Physics.
katex is not defined female students choose Biology.
katex is not defined male students choose Chemistry and katex is not defined choose Biology.
(a) Complete the two way table.
(b) How many female students choose Chemistry?
(c) A student is picked at random. Work out the probability that the student is male and choose physics.
(d) A Biology student is picked at random. Work out the probability that the student is Female.
(9 marks)
(a)
(1)
katex is not defined
(3)
katex is not defined correct
(2)
katex is not defined correct
(1)
(b)
katex is not defined female students choose Chemistry.
(1)
(c)
katex is not defined or katex is not defined
(1)
katex is not defined(1)
(d)
katex is not defined or katex is not defined
(1)
katex is not defined(1)
Learning checklist
You have now learned how to:
- Construct and interpret appropriate tables
- Calculate and interpret conditional probabilities through representation using expected frequencies with two-way tables
The next lessons are
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