Grouped data: estimated mean, modal class and median class

When data is grouped you no longer know the exact values, so you estimate the mean using the middle of each group.

Practise this lesson free (6 questions)

Estimating the mean

Use the midpoint of each class as a stand-in for every value in it.
Multiply each midpoint by its frequency (fx), add them up, and divide by the total frequency.
Time t (min)Frequency fMidpoint xfx
0 < t ≤ 105525
10 < t ≤ 201215180
20 < t ≤ 301825450
30 < t ≤ 40535175
Total40830

Working it out

Estimated mean = 83040 = 20.75 minutes.
It is only an estimate because the real values in each class could be anywhere in the class.

Modal class and median class

The modal class has the highest frequency: 20 < t ≤ 30.
The median is the (40 + 12) = 20.5th value. Cumulative frequencies: 5, 17, 35, 40.
So the median lies in the class 20 < t ≤ 30.
Worked example

Estimate the mean time from the table in the lesson.

  1. Σfx = 830
  2. Σf = 40
  3. 830 ÷ 40 = 20.75

Answer: 20.75 minutes

Key idea

Estimated mean = Σ(midpoint × frequency) ÷ Σfrequency. Modal class: highest frequency. Median class: where the (n + 1)/2th value falls.

Key words
Mean
The total of the values divided by the number of values.
Median
The middle value when the data are in order.
Estimate
An approximate answer, usually found by rounding each number to 1 significant figure.

All GCSE and International GCSE Mathematics key terms and definitions →

Quick check

Try two quick questions

See if it has sunk in. Answers are marked straight away.

Extra practice

2 more questions

Try each one, then tap to check.

  1. What is the midpoint of the class 10 < h ≤ 20?

    Show the answer

    Answer: 15

    (10 + 20) ÷ 2 = 15.

  2. Times t (min): 0 < t ≤ 10, frequency 7; 10 < t ≤ 20, frequency 15; 20 < t ≤ 30, frequency 11; 30 < t ≤ 40, frequency 4. Which is the modal class?

    1. 0 < t ≤ 10
    2. 10 < t ≤ 20
    3. 30 < t ≤ 40
    4. 20 < t ≤ 30
    Show the answer

    Answer: 10 < t ≤ 20

    The modal class has the highest frequency, 15.

Check you have got it

Answer 6 quick questions with instant marking. If you get one wrong, GCSE-ready shows you why and gives you another go. It is free, and you do not need an account.

Practise this lesson free (6 questions)All GCSE and International GCSE Mathematics lessons

Video for this lesson coming soon. Subscribe on YouTube

Sources and references

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

Related lessons