Grouped data and estimated mean | AQA GCSE Maths Notes

Visual model

Use class midpoints for estimated mean

classmidpoint

10\le x<20

15

midpoint times frequency, then divide

Gold-standard guide

20 mins

What you will learn

Estimate averages from grouped frequency tables.

Use a clear step-by-step method for grouped data and estimated mean.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier statistics skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

For grouped data: use the midpoint of each class to represent all values in that class

Step 1

Find midpoints of each class

2.5, 7.5, 12.5, 17.5

Step 2

Multiply each midpoint by its frequency

2.5×3=7.5, 7.5×7=52.5, 12.5×12=150, 17.5×8=140

Step 3

Sum the products and total frequency

Sum of products = 7.5 + 52.5 + 150 + 140 = 350

Watch out

Students use class boundaries instead of midpoints — using 0 or 5 instead of 2.5 for the first class

Midpoint

$midpoint = (lower boundary + upper boundary) / 2.$

Estimated mean

$estimated mean = sum(midpoint \times frequency) / total frequency.$

Worked example

Estimate the mean from this grouped table: [0,5): 3, [5,10): 7, [10,15): 12, [15,20): 8.

Find midpoints of each class: 2.5, 7.5, 12.5, 17.5.

Multiply each midpoint by its frequency: 2.5×3=7.5, 7.5×7=52.5, 12.5×12=150, 17.5×8=140.

Sum the products and total frequency: Sum of products = 7.5 + 52.5 + 150 + 140 = 350. Total frequency = 3+7+12+8 = 30.

Divide: Estimated mean = $35\frac{0}{3}0$ = 11.67 (2 d.p.).

Final answer

Estimated mean ≈ 11.67

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

Estimate the mean from this grouped table: [0,5): 3, [5,10): 7, [10,15): 12, [15,20): 8.

4 marks4 minsgrouped-data-and-estimated-mean-worked

Show solution

Worked solution

1.Find midpoints of each class: 2.5, 7.5, 12.5, 17.5.
2.Multiply each midpoint by its frequency: 2.5×3=7.5, 7.5×7=52.5, 12.5×12=150, 17.5×8=140.
3.Sum the products and total frequency: Sum of products = 7.5 + 52.5 + 150 + 140 = 350. Total frequency = 3+7+12+8 = 30.
4.Divide: Estimated mean = $35\frac{0}{3}0$ = 11.67 (2 d.p.).

Final answer

Estimated mean ≈ 11.67

Mark points

M1: find midpoints of each class
M1: multiply each midpoint by its frequency
M1: sum the products and total frequency
M1: divide
A1: Estimated mean ≈ 11.67

Watch out

5 for the first class. The midpoint is always (lower + upper)/2.Also, dividing the sum of products by the number of classes (not total frequency) is a common error.

Diagnosticrecall

Find the modal class: [0,10): 5, [10,20): 14, [20,30): 9, [30,40): 2.

1 mark2 minsgrouped-data-and-estimated-mean-q1

Show solution

Worked solution

1.Spot the skill: For grouped data: use the midpoint of each class to represent all values in that class.
2.Use the find midpoints of each class stage first, then multiply each midpoint by its frequency.
3.Keep the final answer visible: [10, 20).

Final answer

[10, 20)

Mark points

M1: use the correct for grouped data: use the midpoint of each class to represent all values in that class.multiply each midpoint by its frequency. divide the total by the sum of frequencies.this gives an estimate, not the exact mean.
A1: [10, 20)

Watch out

5 for the first class. The midpoint is always (lower + upper)/2.Also, dividing the sum of products by the number of classes (not total frequency) is a common error.

Easyprocedure

Estimate the mean: [0,4): 6, [4,8): 10, [8,12): 4.

2 marks3 minsgrouped-data-and-estimated-mean-q2

Show solution

Worked solution

1.Spot the skill: For grouped data: use the midpoint of each class to represent all values in that class.
2.Use the multiply each midpoint by its frequency stage first, then sum the products and total frequency.
3.Keep the final answer visible: (2×6 + 6×10 + 10×4)/20 = $11\frac{2}{2}0$ = 5.6.

Final answer

(2×6 + 6×10 + 10×4)/20 = $11\frac{2}{2}0$ = 5.6

Mark points

M1: use the correct for grouped data: use the midpoint of each class to represent all values in that class.multiply each midpoint by its frequency. divide the total by the sum of frequencies.this gives an estimate, not the exact mean.
A1: (2×6 + 6×10 + 10×4)/20 = $11\frac{2}{2}0$ = 5.6

Watch out

5 for the first class. The midpoint is always (lower + upper)/2.Also, dividing the sum of products by the number of classes (not total frequency) is a common error.

Mediumreasoning

Which class contains the median for 30 values with frequencies 8, 12, 10?

3 marks4 minsgrouped-data-and-estimated-mean-q3

Show solution

Worked solution

1.Spot the skill: For grouped data: use the midpoint of each class to represent all values in that class.
2.Use the sum the products and total frequency stage first, then divide.
3.Keep the final answer visible: The 15th and 16th values — both in the second class [frequency 8, then 8+12=20].

Final answer

The 15th and 16th values — both in the second class [frequency 8, then 8+12=20]

Mark points

M1: use the correct for grouped data: use the midpoint of each class to represent all values in that class.multiply each midpoint by its frequency. divide the total by the sum of frequencies.this gives an estimate, not the exact mean.
A1: The 15th and 16th values — both in the second class [frequency 8, then 8+12=20]

Watch out

5 for the first class. The midpoint is always (lower + upper)/2.Also, dividing the sum of products by the number of classes (not total frequency) is a common error.

Hardproblem solving

A survey records [10,20): 5 responses and [20,30): 15. Estimate total mean across both groups.

3 marks5 minsgrouped-data-and-estimated-mean-q4

Show solution

Worked solution

1.Spot the skill: For grouped data: use the midpoint of each class to represent all values in that class.
2.Use the divide stage first, then find midpoints of each class.
3.Keep the final answer visible: Use midpoints 15 and 25: (5×15 + 15×25)/20 = $45\frac{0}{2}0$ = 22.5.

Final answer

Use midpoints 15 and 25: (5×15 + 15×25)/20 = $45\frac{0}{2}0$ = 22.5

Mark points

M1: use the correct for grouped data: use the midpoint of each class to represent all values in that class.multiply each midpoint by its frequency. divide the total by the sum of frequencies.this gives an estimate, not the exact mean.
A1: Use midpoints 15 and 25: (5×15 + 15×25)/20 = $45\frac{0}{2}0$ = 22.5

Watch out

5 for the first class. The midpoint is always (lower + upper)/2.Also, dividing the sum of products by the number of classes (not total frequency) is a common error.

Exam-stylemulti-step

Estimate the mean: [20,30): 4, [30,40): 11, [40,50): 9, [50,60): 6.

4 marks6 minsgrouped-data-and-estimated-mean-q5

Show solution

Worked solution

1.Spot the skill: For grouped data: use the midpoint of each class to represent all values in that class.
2.Use the find midpoints of each class stage first, then multiply each midpoint by its frequency.
3.Keep the final answer visible: (25×4 + 35×11 + 45×9 + 55×6)/30 = $118\frac{0}{3}0$ ≈ 39.3.

Final answer

(25×4 + 35×11 + 45×9 + 55×6)/30 = $118\frac{0}{3}0$ ≈ 39.3

Mark points

M1: use the correct for grouped data: use the midpoint of each class to represent all values in that class.multiply each midpoint by its frequency. divide the total by the sum of frequencies.this gives an estimate, not the exact mean.
A1: (25×4 + 35×11 + 45×9 + 55×6)/30 = $118\frac{0}{3}0$ ≈ 39.3

Watch out

5 for the first class. The midpoint is always (lower + upper)/2.Also, dividing the sum of products by the number of classes (not total frequency) is a common error.

Grade 9 stretchproblem solving

A grouped table has intervals 0 < x ≤ 10 with frequency 4 and 10 < x ≤ 30 with frequency 6. Estimate the mean.

4 marks7 minsgrouped-mean-g9

Show solution

Worked solution

1.Find each midpoint.
2.Multiply midpoint by frequency.
3.Divide the total by total frequency.

Final answer

Mark points

M1: use midpoints 5 and 20
M1: (5 × 4 + 20 × 6) / 10
A1: 14

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint

12 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Grouped data and estimated mean - 2 marksFind the modal class: [0,10): 5, [10,20): 14, [20,30): 9, [30,40): 2.Mark answer

Answer

[10, 20)

2Collecting and sampling data - 2 marksWhy might a questionnaire question be biased?Mark answer

Answer

Leading wording, only offering responses that agree, or not including a 'no' option

3Averages and range - 2 marksThe mean of 5 numbers is 12. Four of them are 8, 14, 10, 15. Find the fifth.Mark answer

Answer

4Charts, tables and diagrams - 3 marksA back-to-back stem-and-leaf shows boys' scores and girls' scores. How do you compare distributions?Mark answer

Answer

Compare medians and ranges for each group

Mastery check

I can explain the method for grouped data and estimated mean.
I can show clear working without skipping key steps.
5 for the first class. The midpoint is always (lower + upper)/2.Also, dividing the sum of products by the number of classes (not total frequency) is a common error.

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