Frequency tables and grouped data are core statistics skills tested across AQA, Edexcel and OCR GCSE Maths papers. At Foundation level you need to read and interpret simple frequency tables; at Higher level you must estimate the mean from grouped data, identify the modal class and find the class interval containing the median. These skills link directly to averages, charts and cumulative frequency, so getting them right unlocks marks across multiple questions. This guide covers everything you need — clear methods, worked examples at both tiers, common pitfalls, and practice questions with full solutions. For a bird's-eye view of every topic, check our complete GCSE Maths topics list.
What Is a Frequency Table?
A frequency table records how often each value or category occurs in a data set. It organises raw data into a clear summary that makes it easier to calculate averages and draw charts.
Simple Frequency Tables
In a simple (ungrouped) frequency table, each row shows a single data value and how many times it appears. For example, recording how many siblings each pupil has:
| Siblings | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Frequency | 3 | 8 | 6 | 2 | 1 |
You can find the mean by adding an extra column for value × frequency, totalling it, and dividing by the total frequency. The mode is the value with the highest frequency (1 sibling, in this case). The median uses cumulative frequency to locate the middle value.
Grouped Frequency Tables
When data covers a wide range of values — such as test marks from 0 to 100 — individual values are grouped into class intervals:
| Marks | 0–19 | 20–39 | 40–59 | 60–79 | 80–100 |
|---|---|---|---|---|---|
| Frequency | 2 | 5 | 12 | 8 | 3 |
With grouped data you lose the individual values, so you can only estimate the mean. You use the midpoint of each class interval as a representative value.
Key Vocabulary
- Class interval — the range each group covers (e.g. 20–39).
- Class width — the difference between the upper and lower boundaries of a class.
- Midpoint — the middle value of a class interval, found by adding the endpoints and dividing by two.
- Modal class — the class interval with the highest frequency.
Step-by-Step Method
Calculating the Mean from a Simple Frequency Table
- Create a new column: value × frequency.
- Add up all the value × frequency results to get the total.
- Add up the frequency column to get the total frequency (n).
- Divide the total by n.
Estimating the Mean from Grouped Data
- Find the midpoint of each class interval.
- Multiply each midpoint by its frequency.
- Add the midpoint × frequency results to get the total.
- Divide by the total frequency.
- State that this is an estimate because you used midpoints, not actual values.
Finding the Median Class
- Calculate the total frequency, n.
- The median is at position (n + 1) ÷ 2 (or n ÷ 2 for large data sets — follow your exam board's convention).
- Use cumulative frequency to identify which class interval contains this position.
Identifying the Modal Class
Simply look for the class interval with the highest frequency. Do not give a single value — state the class interval.
Worked Example 1 — Foundation Level
Question: The table shows the shoe sizes of 20 pupils.
| Shoe size | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|
| Frequency | 2 | 5 | 7 | 4 | 2 |
(a) Find the mode. (b) Calculate the mean shoe size. (c) Find the median shoe size.
Working:
(a) Mode: The highest frequency is 7, which corresponds to shoe size 6. Mode = 6.
(b) Mean:
| Size | Freq | Size × Freq |
|---|---|---|
| 4 | 2 | 8 |
| 5 | 5 | 25 |
| 6 | 7 | 42 |
| 7 | 4 | 28 |
| 8 | 2 | 16 |
Total = 8 + 25 + 42 + 28 + 16 = 119. Total frequency = 20.
Mean = 119 ÷ 20 = 5.95.
(c) Median: n = 20. Position = (20 + 1) ÷ 2 = 10.5th value, so the median is the mean of the 10th and 11th values. Cumulative frequencies: 2, 7, 14, 18, 20. The 10th and 11th values both fall in the shoe size 6 group (positions 8–14). Median = 6.
Worked Example 2 — Higher Level
Question: The grouped frequency table shows the times (in seconds) for 40 pupils to complete a puzzle.
| Time (s) | 10 ≤ t < 20 | 20 ≤ t < 30 | 30 ≤ t < 40 | 40 ≤ t < 50 | 50 ≤ t < 60 |
|---|---|---|---|---|---|
| Frequency | 3 | 9 | 14 | 10 | 4 |
(a) Write down the modal class. (b) Estimate the mean time. (c) Find the class interval containing the median.
Working:
(a) Modal class: The highest frequency is 14, so the modal class is 30 ≤ t < 40.
(b) Estimated mean:
| Class | Midpoint | Freq | Mid × Freq |
|---|---|---|---|
| 10 ≤ t < 20 | 15 | 3 | 45 |
| 20 ≤ t < 30 | 25 | 9 | 225 |
| 30 ≤ t < 40 | 35 | 14 | 490 |
| 40 ≤ t < 50 | 45 | 10 | 450 |
| 50 ≤ t < 60 | 55 | 4 | 220 |
Total = 45 + 225 + 490 + 450 + 220 = 1430. Total frequency = 40.
Estimated mean = 1430 ÷ 40 = 35.75 seconds.
(c) Median class: n = 40, so the median is between the 20th and 21st values. Cumulative frequencies: 3, 12, 26, 36, 40. The 20th and 21st values both fall in the 30 ≤ t < 40 class.
Common Mistakes
- Using end values instead of midpoints when estimating the mean from grouped data. Always add the class boundaries and divide by two.
- Giving a single value for the modal class — write the full class interval, e.g. "30 ≤ t < 40", not "35".
- Forgetting to say "estimate" — when the data is grouped, the mean is an estimate. Examiners expect this word.
- Misreading class boundaries — pay attention to whether the table uses discrete classes (e.g. 0–9, 10–19) or continuous classes (e.g. 0 ≤ x < 10). This affects the midpoint calculation.
- Dividing by the number of classes instead of the total frequency.
Exam Tips
- Set out your working in a table — add columns for midpoints and midpoint × frequency. This keeps your arithmetic organised and earns method marks.
- Label your answer clearly — write "Estimated mean = …" for grouped data.
- Cumulative frequency column — add one when you need to find the median class or draw a cumulative frequency graph. This links to cumulative frequency and box plots.
- Cross-check totals — quickly verify that your frequency total matches the number of data items stated in the question.
- Review key formulas on our GCSE Maths formulas page.
Practice Questions
Question 1 (Foundation): The table shows the number of books read by 25 students last month.
| Books | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Frequency | 4 | 7 | 8 | 4 | 2 |
Calculate the mean number of books and find the mode.
Question 2 (Foundation): Using the same table, find the median.
Question 3 (Higher): The grouped table shows the heights (cm) of 50 plants.
| Height (cm) | 0 ≤ h < 10 | 10 ≤ h < 20 | 20 ≤ h < 30 | 30 ≤ h < 40 |
|---|---|---|---|---|
| Frequency | 6 | 18 | 16 | 10 |
Estimate the mean height and state the modal class.
Question 4 (Higher): Explain why the mean calculated from grouped data is an estimate.
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Related Topics
Summary
Frequency tables organise raw data so you can calculate averages and identify patterns quickly. Simple frequency tables list individual values and their frequencies — you can find exact averages from these. Grouped frequency tables use class intervals for large or continuous data sets — you estimate the mean using midpoints, identify the modal class (not a single value) and locate the median class through cumulative frequency. Always set your working out in a clear table, remember to say "estimate" when the data is grouped, and double-check that your frequency total matches the question. These skills feed directly into drawing histograms, cumulative frequency diagrams and box plots at Higher level.