How to Solve: Averages from Grouped Frequency Tables

How to Solve: Averages from Grouped Frequency Tables

GCSE & A-Level Maths

Introduction

"This one skill can add 5–10 marks to your GCSE or A-Level Maths exam—because grouped frequency tables appear in every paper, and most students lose marks by guessing midpoints or misapplying the formula. Master this, and you’ll solve real-life problems like calculating average salaries, test scores, or even sports performance—all from messy, grouped data."

What You Need To Know First

Before starting, you must understand: 1. Basic averages (mean, median, mode) – how to calculate them from raw data. 2. Class intervals – what they are and how to read them (e.g., "10–20" means values from 10 up to but not including 20). 3. Frequency tables – how to read and interpret them (e.g., "5 people scored between 10 and 20").

Key Vocabulary

Formulas To Know

1. Estimated Mean (Grouped Data)

Formula: [ \text{Estimated Mean} = \frac{\Sigma (f \times x)}{\Sigma f} ] Variables: - ( f ) = frequency of each class - ( x ) = midpoint of each class - ( \Sigma ) = sum of all values

MEMORISE THIS – This is the only formula you need for grouped averages.

Step-by-Step Method

Follow these exact steps for every grouped frequency table question.

1. Read the table carefully.
2. Identify the class intervals and their frequencies.

3. Check if the intervals are inclusive (e.g., 10–19) or exclusive (e.g., 10–<20).

4. Find the midpoint (x) of each class.

5. Add the lower bound + upper bound, then divide by 2.

6. Example: For 20–30 → (20 + 30) ÷ 2 = 25.

7. Multiply each midpoint (x) by its frequency (f).

8. This gives ( f \times x ) for each class.

9. Sum all ( f \times x ) values.

10. This is ( \Sigma (f \times x) ).

11. Sum all frequencies (f).

12. This is ( \Sigma f ) (total number of data points).

13. Divide ( \Sigma (f \times x) ) by ( \Sigma f ).

14. This gives the estimated mean.

15. Write your final answer with units (if any).

16. Example: "The estimated mean is 27.3 kg."

WORKED EXAMPLE (Using the Steps Above)

Question: The table shows the weights of 50 dogs. Calculate the estimated mean weight.

Solution:

1. Read the table.
2. Intervals: 10–15, 15–20, 20–25, 25–30, 30–35.

3. Frequencies: 5, 12, 18, 10, 5.

4. Find midpoints (x).

5. 10–15 → (10 + 15) ÷ 2 = 12.5
6. 15–20 → (15 + 20) ÷ 2 = 17.5
7. 20–25 → (20 + 25) ÷ 2 = 22.5
8. 25–30 → (25 + 30) ÷ 2 = 27.5

9. 30–35 → (30 + 35) ÷ 2 = 32.5

10. Multiply ( f \times x ).

11. 5 × 12.5 = 62.5
12. 12 × 17.5 = 210
13. 18 × 22.5 = 405
14. 10 × 27.5 = 275

15. 5 × 32.5 = 162.5

16. Sum ( f \times x ).

17. 62.5 + 210 + 405 + 275 + 162.5 = 1115

18. Sum frequencies (Σf).

19. 5 + 12 + 18 + 10 + 5 = 50

20. Divide to find the mean.

21. 1115 ÷ 50 = 22.3 kg

Answer: The estimated mean weight is 22.3 kg.

What we did and why: - We used midpoints because we don’t know the exact weights—only ranges. - Multiplying by frequency gives the total weight for each group. - Dividing by total dogs gives the average.

Worked Examples

Example 1 – Basic (GCSE Foundation)

Question: The table shows the number of text messages sent by 40 students in a day. Calculate the estimated mean.

Solution: 1. Midpoints:
- 0–5 → 2.5
- 5–10 → 7.5
- 10–15 → 12.5
- 15–20 → 17.5
- 20–25 → 22.5 2. ( f \times x ):
- 3 × 2.5 = 7.5
- 8 × 7.5 = 60
- 12 × 12.5 = 150
- 10 × 17.5 = 175
- 7 × 22.5 = 157.5 3. ( \Sigma (f \times x) = 7.5 + 60 + 150 + 175 + 157.5 = 550 ) 4. ( \Sigma f = 40 ) 5. Mean = 550 ÷ 40 = 13.75 messages

Answer: The estimated mean is 13.75 messages.

What we did and why: - Midpoints represent the "average" of each group. - ( f \times x ) gives the total messages for each group. - Dividing by 40 gives the average per student.

Example 2 – Medium (GCSE Higher / A-Level)

Question: The table shows the ages of 60 people at a concert. Calculate the estimated mean age.

Solution: 1. Midpoints:
- 10–19 → 14.5
- 20–29 → 24.5
- 30–39 → 34.5
- 40–49 → 44.5
- 50–59 → 54.5 2. ( f \times x ):
- 8 × 14.5 = 116
- 15 × 24.5 = 367.5
- 22 × 34.5 = 759
- 10 × 44.5 = 445
- 5 × 54.5 = 272.5 3. ( \Sigma (f \times x) = 116 + 367.5 + 759 + 445 + 272.5 = 1960 ) 4. ( \Sigma f = 60 ) 5. Mean = 1960 ÷ 60 = 32.67 years

Answer: The estimated mean age is 32.7 years (rounded to 1 d.p.).

What we did and why: - Midpoints are calculated the same way, even with larger intervals. - ( f \times x ) gives the total "age-years" for each group. - Dividing by 60 gives the average age.

Example 3 – Exam-Style (A-Level / GCSE Problem-Solving)

Question: A teacher records the time (in minutes) students spend on homework. The table shows the results for 50 students.

a) Calculate the estimated mean time. b) The teacher claims the mean is 25 minutes. Is this accurate? Explain.

Solution (a): 1. Midpoints:
- 0–10 → 5
- 10–20 → 15
- 20–30 → 25
- 30–40 → 35
- 40–50 → 45 2. ( f \times x ):
- 4 × 5 = 20
- 9 × 15 = 135
- 15 × 25 = 375
- 12 × 35 = 420
- 10 × 45 = 450 3. ( \Sigma (f \times x) = 20 + 135 + 375 + 420 + 450 = 1400 ) 4. ( \Sigma f = 50 ) 5. Mean = 1400 ÷ 50 = 28 minutes

Answer (a): The estimated mean is 28 minutes.

Solution (b): - The teacher’s claim (25 minutes) is lower than the calculated mean (28 minutes). - This could be because they guessed the midpoint or ignored some groups. - The correct mean is 28 minutes, so the claim is not accurate.

What we did and why: - We used ≤ and < to confirm interval boundaries. - The teacher’s mistake was likely underestimating midpoints or miscalculating totals. - Always show working to prove your answer.

Common Mistakes

Exam Traps

1-Minute Recap

"Right, listen up—this is your last-minute checklist for grouped frequency tables:

1. Find midpoints – Add the lower and upper bounds, then divide by 2. No shortcuts!
2. Multiply by frequency – Each midpoint × its frequency gives the total for that group.
3. Sum everything – Add all ( f \times x ) values and all frequencies.
4. Divide – ( \Sigma (f \times x) ) ÷ ( \Sigma f ) = the mean.
5. Check units – Always write "kg," "minutes," etc., in your answer.

Common traps? Unequal intervals, disguised midpoints, forgetting to multiply by frequency. Double-check every step. You’ve got this—go smash that exam!