7.4 Estimating the Mean

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: using midpoints to estimate the mean of grouped data. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Find the midpoint of a class interval
• Estimate the mean from a grouped frequency table
• Explain why the answer is only an estimate
• Use an estimated mean or total to solve problems and test claims

Key words

• Class interval: A group in a grouped frequency table, such as 4 < t ≤ 8.
• Midpoint: The value halfway between the ends of a class: (lower end + upper end) ÷ 2.
• Estimate: A value worked out from the best information you have. It is close to, but not exactly, the true value.
• Total frequency: The total number of items in the table: the sum of the frequency column.

Midpoints

In grouped data you do not know the exact values. We assume every value in a class is at its midpoint.

• Midpoint; 0 < t ≤ 4 2; 10 < x ≤ 20 15; 15 < w ≤ 25 20; 0 to 4 (whole numbers) 2

Worked example: a midpoint

Find the midpoint of 150 < h ≤ 160.
(150 + 160) ÷ 2 = 310 ÷ 2 = 155

Answer: 155

Estimating the mean

Add two columns to the table: midpoint x and x × f. Then estimated mean = total of x × f ÷ total frequency.

• 0 < t ≤ 4; Frequency f 9; Midpoint x 2; x × f 18
• 4 < t ≤ 8; Frequency f 14; Midpoint x 6; x × f 84
• 8 < t ≤ 12; Frequency f 5; Midpoint x 10; x × f 50
• 12 < t ≤ 16; Frequency f 2; Midpoint x 14; x × f 28
• Total; Frequency f 30; Midpoint x ; x × f 180

Worked example: phone calls

x × f: 2 × 9 = 18, 6 × 14 = 84, 10 × 5 = 50, 14 × 2 = 28. Total = 180.
Estimated mean = 180 ÷ 30 = 6

Answer: About 6 minutes

Key fact: Divide by the total frequency (30), never by the number of classes (4).

Why it is only an estimate

We used the midpoints, not the real values. A call in the class 4 estimate.

Worked example: checking an answer makes sense

The estimated mean must lie between the smallest and largest possible values (0 to 16), and it should be near the classes with the biggest frequencies.
6 minutes is inside the class 4 < t ≤ 8, which has the highest frequency.

Answer: 6 minutes is sensible

Using estimates to test a claim

• 0 < x ≤ 10; Frequency f 12; Midpoint x 5; x × f 60
• 10 < x ≤ 20; Frequency f 18; Midpoint x 15; x × f 270
• 20 < x ≤ 30; Frequency f 15; Midpoint x 25; x × f 375
• 30 < x ≤ 40; Frequency f 5; Midpoint x 35; x × f 175
• Total; Frequency f 50; Midpoint x ; x × f 880

Worked example: is the café right?

A café owner says, “On average, customers spend more than £20.” Use the table to test this claim.
Estimated total spent = £880. Estimated mean = 880 ÷ 50 = £17.60.
£17.60 is less than £20.

Answer: The claim is not supported: the estimated mean is £17.60

Common mistakes

• Dividing by the number of classes instead of the total frequency.
• Using the upper end of each class instead of the midpoint.
• Dividing the total of the midpoints by the total frequency (forgetting to multiply by f).
• Giving an answer outside the range of the data, which cannot be right.

Revision questions

Try these, then check your answers below each one.

Question 1. Write down the midpoint of the class 30 < x ≤ 40.

Answer: 35

Working: (30 + 40) ÷ 2 = 35.

Question 2. Write down the midpoint of the class 10 < t ≤ 25.

Answer: 17.5

Working: (10 + 25) ÷ 2 = 17.5.

Question 3. Why is a mean worked out from a grouped frequency table only an estimate? (Choose from: We do not know the exact values in each class, The frequencies might be wrong, Midpoints are always too big, You cannot divide by the total frequency.)

Answer: We do not know the exact values in each class

Working: We assume every value is at the midpoint of its class, but the real values are unknown.

Related lessons

• Previous lesson: 7.3 Types of Average
• Next lesson: 7.5 Sampling
• In this chapter: 7.1 Mean and Range
• In this chapter: 7.2 Mode, Median and Range

Continue in the full course

This lesson is section 7.4 (EGCMF0704) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.