A frequency table shows each value and how many times it occurs. The mode is the value with the highest frequency. To find the mean, add a column of frequency × value, total it, and divide by the total frequency. The median is at position (n + 1) ÷ 2, found by counting through the frequencies, or a running total, until you reach that position.
Frequency table averages
- Total frequency: Σf
- Mean = Σfx ÷ Σf
- Mode: the value with the highest frequency
- Median: position (Σf + 1) ÷ 2, found by counting through the frequencies
- Range = largest value − smallest value (with a frequency above 0)
Reading a frequency table
Each row gives a value x and its frequency f, how many times it occurs. The total frequency, Σf, is the number of data items, not the sum of the values. For a table of books read by a class, Σf is the number of students.
The mode
The mode is the value with the highest frequency. Give the value itself, not the frequency: if 9 students read 2 books and that is the highest frequency, the mode is 2 books, not 9.
The mean
Add a column for f × x, the frequency multiplied by the value. Its total, Σfx, is the total of all the data. Divide by Σf to get the mean. With values 0 to 4 and frequencies 3, 7, 9, 6 and 5, Σf = 30 and Σfx = 63, so the mean is 2.1.
The median
The median is at position (Σf + 1) ÷ 2. With 30 values that is the 15.5th, halfway between the 15th and 16th values. Add up the frequencies as you go, 3, 10, 19 and so on; the 15th and 16th values both lie in the group with running total 19, so both are 2 and the median is 2.
Finding a missing frequency
If the mean is given but one frequency is unknown, write the mean as Σfx ÷ Σf with the unknown included, then form and solve an equation. Multiplying both sides by Σf first keeps the algebra simple.
Worked examples
Example 1. Books read by 30 students: 0 books (3), 1 (7), 2 (9), 3 (6), 4 (5). Find the mode.
- The highest frequency is 9, for 2 books.
Answer: 2 books.
Example 2. Using the same table, find the mean number of books.
- f × x: 0, 7, 18, 18, 20, so Σfx = 63.
- Σf = 30, so the mean = 63 ÷ 30.
Answer: 2.1 books.
Example 3. Using the same table, find the median.
- Median position: (30 + 1) ÷ 2 = 15.5, between the 15th and 16th values.
- Running totals: 3, 10, 19, so the 15th and 16th values are both 2.
Answer: 2 books.
Example 4. Values 1, 2, 3 have frequencies 6, x and 4. The mean is 1.9. Find x.
- Σfx = 6 + 2x + 12 = 18 + 2x and Σf = 10 + x.
- (18 + 2x) ÷ (10 + x) = 1.9, so 18 + 2x = 19 + 1.9x.
- 0.1x = 1.
Answer: x = 10.
Example 5. Using the books table, find the range.
- The largest value with a non-zero frequency is 4; the smallest is 0.
Answer: 4 books.
Common mistakes
- Giving the frequency as the mode. The mode is the value, not how often it occurs.
- Dividing by the number of rows. Divide Σfx by the total frequency Σf.
- Adding the values instead of f × x. The mean needs the f × x column.
- Treating the median position as the median. Count through the frequencies to find the value at that position.
- Forgetting values with zero frequency in the range. Only values that actually occur count.
How frequency tables are examined
Averages from frequency tables are a Foundation staple of Edexcel GCSE (1MA1) and the International GCSE (4MA1), and missing-frequency problems appear at Higher. Questions ask you to:
- write down the mode from a frequency table
- work out the mean, often to 2 decimal places
- find the median using the median position
- find a missing frequency from a given mean
The frequency tables worksheet has worked answers, and the GCSE data revision engine gives more tables to practise on.
For grouped data, see averages from grouped frequency tables.
Frequently asked questions
Which calculation gives the mean of a frequency table?
Multiply each value by its frequency, add these, then divide by the total frequency.
How do you find the median from a frequency table?
Find the position (Σf + 1) ÷ 2, then count through the frequencies to see which value is at that position.
What is the mode in a frequency table?
The value with the highest frequency.