Find the mode, median and mean from a frequency table, and estimate the mean for grouped data using midpoints. It is on Foundation and Higher, aimed at grade 4. Free questions with answers are on this page. Worked examples show each line before you try the questions.
For a frequency table the mode is the group or value with the highest frequency. The median is the 2n+1th value, found by adding frequencies until you pass that position. The mean is ∑f∑fx. For grouped data, x is the midpoint of the class — the mean is then an estimate.
A grouped table cannot give an exact mean or median, because we lost the original numbers. Estimate the median with a running total (or a cumulative-frequency graph later). Check ∑f is the number of people, not ∑fx.
Mode is the value (or class) with the biggest frequency. Estimated mean of grouped data: midpoint × frequency, add, divide by n. Median is the 2n+1th value, found by walking down the cumulative frequency.
Objectives
For the mean, add a column of value × frequency.
Five 2s and one 10 are six numbers, not two rows.
The median is the middle of the ordered list.
Key points
Read the mode straight from the largest frequency.
Find the position of the median using 2n+1 and count through the frequencies.
Calculate the mean from a frequency table with an f×x column.
Use class midpoints to estimate the mean of grouped data.
Worked examples
Example 1
The frequency table shows the number of goals scored in each match: 0 (frequency 4), 1 (frequency 5), 2 (frequency 1). Work out the mean.
The total frequency is 10 and the total of value × frequency is 7.
Mean = 7 ÷ 10 = 0.7.
Answer: 0.7
Common slips
Slip: Value × frequency: 2 × 4 = 0, 1 × 5 = 5, 2 × 1 = 2. That changes a number from the question, so it answers a different question.
Slip: The total frequency is 13 and the total of value × frequency is 7. That changes a number in the working, so the step no longer follows from the one before.
Slip: Mean = 11 ÷ 10 = 0.7. That does not match the working above it.
Example 2
The frequency table shows the number of pets owned by each student: 1 (frequency 5), 2 (frequency 7), 3 (frequency 2). What is the mode?
1 marks
The highest frequency is 7.
That frequency belongs to the value 2.
The mode is 2.
Answer: 2
Common slips
Slip: The highest frequency is 9. That changes a number from the question, so it answers a different question.
Slip: That frequency belongs to the value 5. That changes a number in the working, so the step no longer follows from the one before.
Slip: The mode is 6. That does not match the working above it.
Example 3
The frequency table shows the number of siblings of each student: 1 (frequency 3), 2 (frequency 6), 3 (frequency 1), 4 (frequency 3). What is the median?
3 marks
The total frequency is 13, so the median is the 7th value.
Running total of the frequencies: 1 up to 3, 2 up to 9, 3 up to 10, 4 up to 13.
The 7th value is 2.
Answer: 2
Common slips
Slip: The total frequency is 15, so the median is the 7th value. That changes a number from the question, so it answers a different question.
Slip: Running total of the frequencies: 4 up to 3, 2 up to 9, 3 up to 10, 4 up to 13. That changes a number in the working, so the step no longer follows from the one before.
Slip: The 11th value is 2. That does not match the working above it.
Questions and answers
GCSE Maths averages from frequency tables questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
The frequency table shows the number of books read by each student this month: 0 (frequency 4), 1 (frequency 6), 2 (frequency 6), 3 (frequency 4). Work out the mean.
The frequency table shows the number of pets owned by each student: 0 (frequency 3), 1 (frequency 5), 2 (frequency 5), 3 (frequency 6), 4 (frequency 2). Work out the range.
Show answer
Answer: 4
The largest value is 4 and the smallest value is 0.
Range = 4 − 0.
M1 — The largest value is 4 and the smallest value is 0.
A1 — Answer 4
Question 3 Medium2 marks
The frequency table shows the number of goals scored in each match: 1 (frequency 7), 2 (frequency 5), 3 (frequency 3), 4 (frequency 2). What is the mode?
Show answer
Answer: 1
The highest frequency is 7.
That frequency belongs to the value 1.
M1 — The highest frequency is 7.
A1 — Answer 1
Question 4 Medium2 marks
The frequency table shows the number of siblings of each student: 1 (frequency 4), 2 (frequency 5), 3 (frequency 2), 4 (frequency 5). Work out the mean.
The frequency table shows the number of pets owned by each student: 1 (frequency 2), 2 (frequency 4), 3 (frequency 1), 4 (frequency 5), 5 (frequency 3). What is the median?
Show answer
Answer: 4
The total frequency is 15, so the median is the 8th value.
Running total of the frequencies: 1 up to 2, 2 up to 6, 3 up to 7, 4 up to 12, 5 up to 15.
M1 — The total frequency is 15, so the median is the 8th value.
A1 — Answer 4
Question 6 Hard2 marks
The frequency table shows the number of books read by each student this month: 1 (frequency 1), 2 (frequency 6), 3 (frequency 2), 4 (frequency 4), 5 (frequency 2). What is the mode?
Show answer
Answer: 2
The highest frequency is 6.
That frequency belongs to the value 2.
M1 — The highest frequency is 6.
A1 — Answer 2
Question 7 Exam-style3 marks
The frequency table shows the number of siblings of each student: 0 (frequency 3), 1 (frequency 4), 2 (frequency 7), 3 (frequency 6). Work out the mean.