When data is continuous or spans a wide range, a grouped frequency table organises it into class intervals. This makes patterns easier to spot. From the table you can identify the modal class, estimate the mean, and describe the spread — all key skills for KS3 statistics.
What is a grouped frequency table?
A frequency table records how often each value appears in a data set. When the data is continuous (such as heights, times, or temperatures) or when there are many different values spread over a wide range, listing every single value individually creates an unwieldy table. Instead, values are gathered into bands called class intervals, and the table records how many data values fall into each band.
For example, rather than recording every pupil's exact height to the nearest millimetre, a teacher might record:
- how many pupils have heights in the range 140 cm to 150 cm
- how many are in the range 150 cm to 160 cm
- and so on up to the tallest pupil.
Each row in the completed table shows one class interval alongside its frequency — the count of data values that fall within that interval. The table is compact, easy to read, and still contains enough information to calculate key statistics.
How do I read class intervals?
Class intervals in KS3 maths are written using inequality notation. For example:
140 ≤ h < 150
This means: heights h that are at least 140 cm but strictly less than 150 cm. A value of exactly 140 is included in this group; a value of exactly 150 is excluded and goes into the next group (150 ≤ h < 160). This convention removes any ambiguity about where boundary values belong.
The width of a class interval is the difference between its upper and lower boundary: 150 − 140 = 10 cm in this case. In a well-designed table all class intervals usually have equal width, which keeps analysis consistent and straightforward.
What is the modal class?
The modal class is the class interval with the highest frequency. It plays the same role as the mode does for ungrouped data — it tells you where the data clusters most densely.
In the table below — showing the heights of 20 students — the modal class is 150 ≤ h < 160 because that row has the highest frequency of 7.
| Height (cm) | Frequency |
|---|---|
| 140 ≤ h < 150 | 3 |
| 150 ≤ h < 160 | 7 |
| 160 ≤ h < 170 | 6 |
| 170 ≤ h < 180 | 4 |
An important point: you cannot give a single modal value (such as "the mode is 155 cm") from a grouped frequency table. You can only state the class: "the modal class is 150 ≤ h < 160". The table does not tell you which value inside that class appears most often.
How do I estimate the mean from a grouped frequency table?
Because you do not know the individual values, you assume that every data value in a class interval is located at its midpoint. The midpoint of 140 ≤ h < 150 is (140 + 150) ÷ 2 = 145 cm. This assumption introduces a small error, which is why the result is called an estimated mean rather than the exact mean.
The formula is:
Estimated mean = Σ(midpoint × frequency) ÷ Σfrequency
Worked example — heights of 20 students
Add a midpoint column and a midpoint × frequency column to the table.
| Height (cm) | Frequency | Midpoint | Midpoint × Frequency |
|---|---|---|---|
| 140 ≤ h < 150 | 3 | 145 | 435 |
| 150 ≤ h < 160 | 7 | 155 | 1085 |
| 160 ≤ h < 170 | 6 | 165 | 990 |
| 170 ≤ h < 180 | 4 | 175 | 700 |
| Totals | 20 | — | 3210 |
Step-by-step method:
- Find the midpoint of each class: (lower boundary + upper boundary) ÷ 2.
- Multiply each midpoint by its frequency.
- Sum the frequency column: 3 + 7 + 6 + 4 = 20.
- Sum the midpoint × frequency column: 435 + 1085 + 990 + 700 = 3210.
- Divide: 3210 ÷ 20 = 160.5 cm (estimated mean).
Check the arithmetic: 435 + 1085 = 1520; 1520 + 990 = 2510; 2510 + 700 = 3210 ✓
Why is the mean only an estimate?
The grouped table does not record individual heights — only the count within each 10-cm band. Using the midpoint of each band is a reasonable but imperfect assumption. Unless every data value in a class happens to sit exactly at the midpoint, the estimated mean will differ slightly from the true mean that you would obtain from the raw data.
This distinction matters in exams: write "estimated mean" rather than simply "mean" to show you understand that grouped data can only yield an approximation. The same logic applies to the median — from a grouped table you can only identify which class the median lies in, not its exact value.
Worked example 2 — finding a missing frequency
Twenty-five students sat a test. The grouped frequency table below has one frequency missing. Find the value of f.
| Score | Frequency |
|---|---|
| 0 ≤ s < 20 | 2 |
| 20 ≤ s < 40 | 5 |
| 40 ≤ s < 60 | f |
| 60 ≤ s < 80 | 8 |
| 80 ≤ s ≤ 100 | 3 |
The frequencies must sum to the total number of students:
2 + 5 + f + 8 + 3 = 25 → 18 + f = 25 → f = 7
What are the most common mistakes with grouped frequency tables?
-
Using a class boundary instead of the midpoint. A very common slip is to write 150 × 7 (using the boundary) instead of 155 × 7 (using the midpoint). Always calculate the midpoint as (lower + upper) ÷ 2 before multiplying by frequency.
-
Stating an exact mode or median. From a grouped table you can name the modal class and the class containing the median, but never a single exact value. Write "the modal class is 150 ≤ h < 160", not "the mode is 155 cm".
-
Not checking that frequencies sum to the stated total. Whenever a table includes a missing frequency, write the sum equation first. It is almost always the fastest route to the answer and prevents sign errors.
-
Writing "mean" instead of "estimated mean". Many mark schemes award the final mark only when the word "estimated" appears. Form the habit of writing "estimated mean = …" every time the data is grouped.
Frequently asked questions
Can I find the exact median from a grouped frequency table?
No — you can only identify which class interval contains the median. For 20 data values the median lies between the 10th and 11th values; cumulating the frequencies shows that both fall in the 150 ≤ h < 160 class. To approximate its position within that class you would use linear interpolation, but at KS3 you are expected only to state which class the median lies in, not to interpolate.
What is the difference between the modal class and the mode?
The mode is a single data value that appears most often, and can be identified exactly from a list or an ungrouped frequency table. The modal class is the class interval with the highest frequency in a grouped frequency table. Because individual values are hidden within classes, you cannot name a single modal value — only the interval that contains the most data.
Why do class intervals use ≤ and < rather than "from … to …"?
The inequality notation removes any ambiguity at boundary values. Writing "140 to 150" leaves it unclear whether a height of exactly 150 belongs in that group or the next. With 140 ≤ h < 150, a value of exactly 150 is excluded (< means strictly less than) and goes into 150 ≤ h < 160. Every possible measurement is allocated to exactly one class with no overlap and no gap.
Do class intervals always have to be the same width?
At KS3 they are almost always equal in width, which keeps analysis — particularly the estimated mean — straightforward. Unequal class widths appear in GCSE Higher histograms, where frequency density (frequency ÷ class width) must be used on the vertical axis rather than frequency. Unless told otherwise at KS3, assume all class intervals are the same width.
For guided KS3 statistics practice with grouped data, modal class, and estimated mean, try Professor Pi at aitutors.me.