Calculating the mean from a frequency table is one of the most common statistics questions on both Foundation and Higher GCSE papers. Whether the data is ungrouped or grouped, the core method is the same: multiply each value (or midpoint) by its frequency, add them up, and divide by the total frequency. This guide covers both types, along with modal class and how to find the class containing the median.
What Is the Mean from a Frequency Table?
When data is presented in a frequency table, you cannot simply add all the values and divide by how many there are — because each value appears multiple times. Instead, you use the frequencies to weight each value before averaging.
For ungrouped data, each row shows a specific data value and how many times it appears. You multiply each value by its frequency (giving fx), sum the fx column, and divide by the total frequency.
For grouped data, the individual values are unknown because they are collected into class intervals (e.g., 10-19, 20-29). You use the midpoint of each class as a representative value. The result is called an estimated mean because you do not know the exact data values. The modal class is the class interval with the highest frequency, and the class containing the median is found by counting through the cumulative frequencies.
Key Formulas
Step-by-Step Method
- If grouped, find the midpoint of each class interval.
- Multiply each value (or midpoint) by its frequency to create the fx column.
- Add up all the fx values to get Σfx.
- Add up all the frequencies to get Σf.
- Divide Σfx by Σf to get the mean.
Worked Example 1 — Foundation Level
Question: The table shows the number of pets owned by 30 students. Find the mean.
| Pets | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Frequency | 5 | 8 | 10 | 4 | 3 |
Working: fx values: 0×5=0, 1×8=8, 2×10=20, 3×4=12, 4×3=12. Σfx = 0 + 8 + 20 + 12 + 12 = 52. Σf = 5 + 8 + 10 + 4 + 3 = 30. Mean = 52 ÷ 30 = 1.73 (to 3 s.f.).
Answer: The mean number of pets is 1.73.
Worked Example 2 — Higher Level
Question: The grouped frequency table shows test scores for 50 students. Find the estimated mean.
| Score | 0-19 | 20-39 | 40-59 | 60-79 | 80-100 |
|---|---|---|---|---|---|
| Frequency | 3 | 8 | 15 | 18 | 6 |
Working: Midpoints: 9.5, 29.5, 49.5, 69.5, 90. fx values: 9.5×3=28.5, 29.5×8=236, 49.5×15=742.5, 69.5×18=1251, 90×6=540. Σfx = 28.5 + 236 + 742.5 + 1251 + 540 = 2798. Σf = 3 + 8 + 15 + 18 + 6 = 50. Estimated mean = 2798 ÷ 50 = 55.96.
Answer: The estimated mean score is 55.96.
Worked Example 3 — Exam Style
Question: Using the table in Worked Example 2, state the modal class and find the class interval containing the median.
Working: The modal class has the highest frequency: 60-79 (frequency 18). Total frequency = 50, so the median is between the 25th and 26th values. Cumulative frequencies: 3, 11, 26, 44, 50. The 25th and 26th values both fall in the 40-59 class (cumulative frequency reaches 26 at the end of this class).
Answer: Modal class is 60-79. The median lies in the 40-59 class.
Common Mistakes
- Using class boundaries instead of midpoints. For grouped data, always calculate the midpoint of each class. Do not use the lower or upper boundary.
- Dividing by the number of classes instead of the total frequency. The denominator is Σf (total number of data items), not the number of rows in the table.
- Forgetting the last class has a different width. In the worked example above, 80-100 has a midpoint of 90 (not 89.5) because the class includes 100. Read class boundaries carefully.
Exam Tips
- Add extra columns to the table for midpoints and fx — examiners expect to see this working and award marks for it.
- For the class containing the median, use cumulative frequency. The median position is at the (n + 1)/2 th value (or n/2 th for grouped data).
- Always label your answer "estimated mean" when working with grouped data to show you understand it is an approximation.
Practice Questions
Q1 (Foundation): Find the mean from this frequency table.
| Score | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Frequency | 4 | 7 | 9 | 5 | 5 |
Q2 (Foundation): The modal value in Q1 is the value with the highest frequency. State the mode.
Q3 (Higher): Heights of 40 plants are recorded below. Calculate the estimated mean.
| Height (cm) | 0-9 | 10-19 | 20-29 | 30-39 |
|---|---|---|---|---|
| Frequency | 6 | 14 | 12 | 8 |
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Related Topics
Summary
- For ungrouped frequency tables: mean = Σfx ÷ Σf.
- For grouped data: use midpoints in place of exact values to get the estimated mean.
- The modal class is the class with the highest frequency.
- Find the class containing the median by building cumulative frequencies and counting to the middle value.
- Always show your fx column and label grouped results as an "estimated mean".
Test your understanding
5 quick MCQs to identify any misconceptions on this topic.
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