Pass an Exam 
Hi, I’m Ari. We can start frequency tables from the beginning, work through an example together, or practise a question. Tell me which step is confusing.
GCSE Maths · Statistics
Frequency tables GCSE Questions and Worked Answers
Frequency tells you how often a value or category occurs. Add frequencies to count observations; multiply value by frequency to recover a numerical total. Grouped data usually gives estimates rather than exact individual values.
Free AI tutor · GCSE frequency tables
Practise GCSE frequency tables for free with an AI tutor
Ask Ari for an explanation or work through an original exam-style question together.
AriYour maths coach
Chat cost ≈ $0.000000
Ari is an AI tutor and can make mistakes. Use the worked answers below to check important results.
Start with the meaning
What you need to know about frequency tables
Suppose players score 0, 0, 1, 1, 1, 2, 2, 2, 2 and 3 goals. Rather than repeat every number, record that zero occurs twice, one three times, two four times and three once. These occurrence counts are called frequencies.
See the idea first
A row represents repeated observations
In the table, x means the number of goals and f the number of players with that score. Four players scoring two goals contribute 2 × 4 = 8 goals. Add the frequency column to count people; add the x × f column to count goals.
| Goals x | Players f | Total goals x × f |
|---|---|---|
| 0 | 2 | 0 |
| 1 | 3 | 3 |
| 2 | 4 | 8 |
| 3 | 1 | 3 |
From problem to method
Typical problems you need to be able to solve
These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.
Find a total and mean
What the problem asks: Use the goals table to find the number of players and mean goals.
How to solve it: There are 2 + 3 + 4 + 1 = 10 players. Goals total 0 + 3 + 8 + 3 = 14. Mean = 14 ÷ 10 = 1.4.
Find the median
What the problem asks: Use the same table to find the median number of goals.
How to solve it: Ten players give middle positions 5 and 6. Positions 3–5 score 1, while positions 6–9 score 2. Median = (1 + 2) ÷ 2 = 1.5.
Estimate from groups
What the problem asks: Times are grouped as 0 < t ≤ 10, 10 < t ≤ 20 and 20 < t ≤ 30 seconds. How can a mean be estimated?
How to solve it: Use each interval's midpoint as a representative value: 5, 15 and 25. Multiply midpoint by frequency, add and divide by total frequency. Exact times have been lost, so this is an estimate.
A reliable routine
Recover a mean from repeated values
For a numerical frequency table, x × f reconstructs the total contribution of a row because it replaces adding x repeatedly. The compact notation Σ means ‘add all rows’: mean = Σfx ÷ Σf.
- Read value and frequency headings; count total frequency.
- Multiply each numerical value by its frequency. For grouped data, first use class midpoints.
- Add these products and divide by the total frequency, not the number of rows.
- For median instead, accumulate frequencies until you locate the middle position or positions.
Check: A class such as 10 < t ≤ 20 includes 20 but excludes 10. A categorical label such as colour has no meaningful numerical mean.
Fully worked
Frequency tables GCSE worked examples
Each solution explains the clue, why the method fits and how to check the result.
Example 1
Make a frequency table
Question
Summarise the scores 1, 3, 1, 2, 3, 3, 2, 1 in a frequency table.
Count once per observation: score 1 occurs 3 times; score 2 occurs 2 times; score 3 occurs 3 times. Check total frequency: 3 + 2 + 3 = 8.
| Score | Frequency |
|---|---|
| 1 | 3 |
| 2 | 2 |
| 3 | 3 |
Example 2
Mean from frequencies
Question
Find the mean number of goals from this table.
| Goals x | Players f | Total goals x × f |
|---|---|---|
| 0 | 2 | 0 |
| 1 | 3 | 3 |
| 2 | 4 | 8 |
| 3 | 1 | 3 |
Total frequency is .
The zero-goal players still count in the denominator.
Example 3
Median and mode
Question
Find the median and mode of the goals table.
| Goals x | Players f | Total goals x × f |
|---|---|---|
| 0 | 2 | 0 |
| 1 | 3 | 3 |
| 2 | 4 | 8 |
| 3 | 1 | 3 |
The cumulative counts are 2, 5, 9, 10. Position 5 has value 1 and position 6 value 2.
The greatest frequency is 4, corresponding to 2 goals, so the mode is 2, not 4.
Example 4
Grouped mean estimate
Question
Estimate the mean time using this grouped table.
| Time t (s) | Frequency |
|---|---|
| 0 < t ≤ 10 | 4 |
| 10 < t ≤ 20 | 10 |
| 20 < t ≤ 30 | 6 |
Midpoints are 5, 15 and 25 s.
We do not know individual times, so the mean is not exact.
Example 5
Median class
Question
Identify the class containing the median in the grouped times table.
| Time t (s) | Frequency |
|---|---|
| 0 < t ≤ 10 | 4 |
| 10 < t ≤ 20 | 10 |
| 20 < t ≤ 30 | 6 |
There are 20 times, so middle positions are 10 and 11. Cumulative frequencies are 4, 14, 20: both positions lie in 10 < t ≤ 20 seconds. The exact median cannot be recovered from these groups alone.
Example 6
Missing frequency
Question
Scores 1, 2 and 3 have frequencies 2, k and 4. Their mean is 2.2. Find k.
Total frequency is k + 6; total score is 2k + 14.
Check: total 22 divided by 10 observations is 2.2.
10 original questions · total 20 marks
Frequency tables GCSE exam-style questions
Try each question before opening its fully worked answer. The difficulty rises through the set.
Before you startAllow about 25 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed
Read a row
A table records ‘2 pets, frequency 6’. How many pets do those six households have in total?
Show worked answer
Six households each have two pets:
Count observations
Frequencies are 3, 7, 5 and 2. Find the total number of observations.
Show worked answer
Add frequencies, not category labels.
Mean
Values 1, 2 and 4 have frequencies 2, 5 and 3. Find the mean.
Show worked answer
Mode
Scores 0, 1, 2 and 3 have frequencies 2, 8, 5 and 1. Find the mode.
Show worked answer
The greatest frequency is 8, which belongs to score 1. The mode is the score, not its frequency.
Median position
Values 2, 5 and 7 have frequencies 3, 4 and 2. Find the median.
Show worked answer
There are nine observations, so locate position 5. The first three values are 2, positions 4–7 are 5, and the last two are 7. Median = 5.
Class boundaries
Which class contains 20: 10 < x ≤ 20 or 20 < x ≤ 30? Explain.
Show worked answer
The first class includes 20 through ≤. The second excludes 20 through <. Each observation belongs in exactly one class.
Grouped estimate
Classes 0 < x ≤ 10 and 10 < x ≤ 20 have frequencies 3 and 7. Estimate the mean.
Show worked answer
Midpoints are 5 and 15.
It is an estimate because actual values within each class are unknown.
Modal class
The equal-width classes 0 < t ≤ 5, 5 < t ≤ 10 and 10 < t ≤ 15 have frequencies 4, 9 and 6. Find the modal class.
Show worked answer
The greatest frequency is 9, so the modal class is 5 < t ≤ 10, not the number 9.
A category mean?
A table counts red, green and blue cars. Can it give a meaningful mean colour? Explain.
Show worked answer
No. Colour categories are not numerical measurements. You can find the most frequent colour, the mode, or each colour's proportion.
Combine counts
One class records 5 pupils with 0 siblings and 15 with 1 sibling. Another records 3 with 0 and 7 with 1. Find the combined mean number of siblings.
Show worked answer
Combined frequencies: zero has 8, one has 22. There are 30 pupils and 22 siblings counted.
This is about 0.733 siblings per pupil.
Examiner-style feedback
Common frequency tables mistakes
The observations are counted by total frequency, not by the number of distinct values.
Report the value or category whose frequency is largest.
Midpoints represent unknown values; state that the result is an estimate.
30-second recap
Choose, carry out, check
Translate the wording before you reach for a rule.
- Read what each column counts.
- Multiply values by their frequencies.
- Divide a total by the observation count.
- Treat grouped values as estimates.
Quick answers
Frequency tables FAQ
What does Σ mean?
It means add all the entries indicated. Σf is total frequency; Σfx is the sum of value × frequency products.
Can I always find an exact grouped median?
No. Groups locate a median class, but do not reveal the individual middle observations.
Content standards
Curriculum and rights review
S2/S4 numerical and categorical frequency tables, grouped estimates and median classes across tiers. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.
Official specification references