Frequency Tables and Two-Way Tables

This lesson covers how to organise raw data into frequency tables, grouped frequency tables, and two-way tables — all essential skills for the AQA GCSE Mathematics Statistics unit. Being able to construct, read, and interpret these tables is fundamental to almost every statistics question in the exam.

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Tally Charts and Frequency Tables

A tally chart records data using tally marks as it is collected. Every fifth tally mark is drawn diagonally across the previous four to make groups of five, which makes counting easier.

A frequency table summarises data by showing how many times each value or category occurs.

Worked Example

Thirty students were asked how many books they read last month. The raw data is:

2, 0, 3, 1, 1, 4, 2, 2, 0, 3, 5, 1, 2, 3, 1, 0, 2, 4, 1, 3, 2, 1, 0, 2, 3, 1, 4, 2, 1, 3

Books Read	Tally	Frequency
0	IIII	4
1	IIII III	8
2	IIII III	8
3	IIII I	6
4	III	3
5	I	1
Total		30

Exam Tip: Always check that the total frequency matches the number of data values given. If they do not match, you have made a counting error.

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Grouped Frequency Tables

When data has a wide range of values (especially continuous data), we organise it into class intervals (groups).

Key Rules for Class Intervals

• Class intervals must not overlap — each data value must fit into exactly one group.
• Class intervals should not have gaps between them (for continuous data).
• All intervals should ideally be the same width unless there is a good reason otherwise.

Worked Example — Discrete Data

The ages of 25 people at a cinema:

Age	Frequency
10–19	6
20–29	8
30–39	5
40–49	4
50–59	2
Total	25

Here the class intervals do not overlap: a person aged 19 goes in the 10–19 group, a person aged 20 goes in the 20–29 group.

Worked Example — Continuous Data

The heights of 30 plants in centimetres:

Height (h cm)	Frequency
0 < h ≤ 10	3
10 < h ≤ 20	7
20 < h ≤ 30	11
30 < h ≤ 40	6
40 < h ≤ 50	3
Total	30

Notice the use of inequality notation for continuous data. A plant measuring exactly 10 cm would go in the first class (0 < h ≤ 10), not the second.

Exam Tip: Pay close attention to the inequality signs. The notation 10 < h ≤ 20 means h is greater than 10 and less than or equal to 20. A value of exactly 10 would NOT be included in this class.

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Reading Frequency Tables

You should be able to use a frequency table to answer questions about the data.

Common Questions

Given the books-read frequency table above:

1. How many students read exactly 2 books? Read directly from the table: 8 students.

2. How many students read fewer than 3 books? Add the frequencies for 0, 1, and 2: 4 + 8 + 8 = 20 students.

3. How many students read 3 or more books? Add the frequencies for 3, 4, and 5: 6 + 3 + 1 = 10 students.

4. What fraction of students read more than 3 books? Frequency for 4 and 5: 3 + 1 = 4. Fraction = 4/30 = 2/15.

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Two-Way Tables

A two-way table displays data that has been classified by two categories simultaneously. They are laid out as a grid with row totals and column totals.

Worked Example

A group of 60 students were asked whether they prefer football, tennis, or swimming. The results are split by gender.

	Football	Tennis	Swimming	Total
Boys	15	8	9	32
Girls	7	12	9	28
Total	22	20	18	60

From this table we can see:

• 15 boys prefer football.
• 12 girls prefer tennis.
• 22 students in total prefer football.
• There are 32 boys and 28 girls.
• The grand total is 60.

Completing a Two-Way Table

Often you will be given a partially completed two-way table and asked to fill in the missing values.

Worked Example

	Cat	Dog	Other	Total
Male	12	?	5	30
Female	?	11	?	?
Total	20	26	14	60

Step 1: Male dogs: 30 - 12 - 5 = 13

Step 2: Female total: 60 - 30 = 30

Step 3: Female cats: 20 - 12 = 8

Step 4: Female dogs: 26 - 13 = 13 (or alternatively 30 - 8 - ? requires finding Other first)

Step 5: Female other: 14 - 5 = 9

Check: Female row: 8 + 13 + 9 = 30. Correct.

Completed table:

	Cat	Dog	Other	Total
Male	12	13	5	30
Female	8	13	9	30
Total	20	26	14	60

Exam Tip: When completing a two-way table, always start with the row or column where you have the most information (i.e. only one value is missing). Use the totals to work backwards. Always verify your answers by checking that every row and column adds up correctly.

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Using Two-Way Tables for Probability

Two-way tables can be used to calculate probabilities.

Using the sports table above: