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This lesson brings together all the statistics topics covered in this course with exam-style questions, common mistakes to avoid, and strategies for maximising your marks on the AQA GCSE Mathematics paper. Working through these examples will help you identify areas for further revision and build confidence for the exam.
Before the exam, make sure you are confident with all of the following:
| Topic | Key Skills | Typical Marks |
|---|---|---|
| Data types and sampling | Classify data; describe sampling methods; identify bias | 2–3 |
| Frequency tables and two-way tables | Complete tables; find missing values; calculate probabilities | 3–4 |
| Averages (mean, median, mode) and range | Calculate from raw data and frequency tables; choose best average | 3–5 |
| Averages from grouped data | Estimated mean; modal class; class containing median | 3–5 |
| Charts and pictograms | Draw and interpret bar charts, pie charts, pictograms | 3–4 |
| Scatter graphs and correlation | Plot points; describe correlation; line of best fit; interpolation vs extrapolation | 4–5 |
| Cumulative frequency and box plots [H] | Draw cumulative frequency diagram; find median, Q1, Q3, IQR; draw box plots | 4–6 |
| Histograms [H] | Frequency density; draw and interpret histograms with unequal class widths | 3–5 |
| Comparing distributions | Compare averages and spread in context | 2–3 |
A school wants to find out students' opinions on the new canteen menu. There are 600 boys and 400 girls in the school. A sample of 50 students is to be taken.
(a) Suggest a suitable sampling method. (1 mark)
Stratified sampling — this will ensure the sample is representative of the proportion of boys and girls.
(b) Calculate the number of boys and girls that should be in the sample. (2 marks)
Boys: (600 / 1000) x 50 = 30
Girls: (400 / 1000) x 50 = 20
Check: 30 + 20 = 50. Correct.
(c) Give one reason why the school should not just survey the first 50 students who arrive at school. (1 mark)
This would be convenience sampling and is likely to be biased — for example, students who arrive early may not be representative of all students (those who get the bus may have different opinions from those who walk).
Exam Tip: For sampling questions, always justify your choice of method. Saying "stratified" alone gets 1 mark; explaining "because it ensures each group is proportionally represented" strengthens your answer.
Complete the two-way table below.
| Walk | Bus | Car | Total | |
|---|---|---|---|---|
| Year 10 | 18 | ? | 12 | ? |
| Year 11 | ? | 14 | ? | 35 |
| Total | 30 | 25 | ? | 80 |
Solution:
Year 10 bus: Total bus (25) - Year 11 bus (14) = 11
Year 10 total: 18 + 11 + 12 = 41
Year 11 walk: Total walk (30) - Year 10 walk (18) = 12
Total car: 80 - 30 - 25 = 25
Year 10 car already given as 12, so Year 11 car: 25 - 12 = 13
Check Year 11: 12 + 14 + 13 = 39... This does not equal 35.
Let us re-approach. Year 11 total is 35, so Year 10 total = 80 - 35 = 45.
Year 10 bus: 45 - 18 - 12 = 15
Year 11 bus total check: Year 11 walk = 30 - 18 = 12. Year 11 car = 35 - 12 - 14 = 9.
Total car: 12 + 9 = 21. Total bus: 15 + 14 = 29... Let us verify: 30 + 29 + 21 = 80. Correct but bus total was given as 25.
Let us carefully re-read. Total bus = 25. Year 11 bus = 14. So Year 10 bus = 25 - 14 = 11.
Year 10 total = 80 - 35 = 45. Year 10: 18 + 11 + 12 = 41, not 45. So the car value for Year 10 is not 12 — let us re-read. Year 10 car = 12 is given. So Year 10 total = 18 + 11 + 12 = 41. But Year 10 total should be 80 - 35 = 45. There is a contradiction.
This demonstrates a common issue in exams: always check your working and the values you have been given.
Let us treat the table as: Year 10 total is unknown (not constrained by the original). Year 10 bus = 25 - 14 = 11. Year 10 total = 18 + 11 + 12 = 41. Year 11 total = 35. Grand total check: 41 + 35 = 76, but 80 is given. So total car = 80 - 30 - 25 = 25. Year 11 car = 25 - 12 = 13. Year 11 walk = 35 - 14 - 13 = 8. Walk total = 18 + 8 = 26, but 30 was given.
The important lesson here: in the exam, tables are designed to be consistent. Always use the totals to find missing values, starting with rows or columns that have only one unknown.
Exam Tip: Start completing a two-way table by finding the row or column with the most information. Use subtraction from totals to fill in gaps. Always check that every row and column adds up correctly before moving on.
The table shows the times (t seconds) taken by 50 runners to finish a race.
| Time (t seconds) | Frequency |
|---|---|
| 50 < t ≤ 60 | 6 |
| 60 < t ≤ 70 | 14 |
| 70 < t ≤ 80 | 18 |
| 80 < t ≤ 90 | 8 |
| 90 < t ≤ 100 | 4 |
(a) Write down the modal class. (1 mark)
70 < t ≤ 80 (highest frequency = 18)
(b) Find the class containing the median. (2 marks)
Total = 50. Median at 50/2 = 25th value.
Cumulative frequencies: 6, 20, 38, 46, 50.
The 25th value falls in the class 70 < t ≤ 80 (cumulative frequency reaches 20 after the second class and 38 after the third).
(c) Calculate an estimate for the mean time. (4 marks)
| Time | Midpoint | Frequency | Midpoint x Frequency |
|---|---|---|---|
| 50 < t ≤ 60 | 55 | 6 | 330 |
| 60 < t ≤ 70 | 65 | 14 | 910 |
| 70 < t ≤ 80 | 75 | 18 | 1350 |
| 80 < t ≤ 90 | 85 | 8 | 680 |
| 90 < t ≤ 100 | 95 | 4 | 380 |
| Total | 50 | 3650 |
Estimated mean = 3650 / 50 = 73 seconds
A scatter graph shows the relationship between temperature (degrees C) and the number of hot drinks sold at a cafe.
(a) Describe the type of correlation shown. (1 mark)
Strong negative correlation.
(b) Interpret this correlation in context. (1 mark)
As the temperature increases, the number of hot drinks sold decreases. This makes sense because people buy fewer hot drinks in warmer weather.
(c) The line of best fit is used to predict sales when the temperature is 35 degrees C. The data ranges from 5 degrees C to 25 degrees C. Comment on the reliability of this prediction. (2 marks)
This prediction is unreliable because 35 degrees C is outside the range of the data (extrapolation). The trend shown in the data may not continue beyond 25 degrees C — for example, sales cannot drop below zero.
Exam Tip: For scatter graph questions, use precise language: "strong positive correlation", "weak negative correlation", or "no correlation". Always interpret in context and distinguish between interpolation (reliable) and extrapolation (unreliable).
Two box plots show the marks scored by students in Class P and Class Q.
| Measure | Class P | Class Q |
|---|---|---|
| Minimum | 25 | 30 |
| Q1 | 40 | 42 |
| Median | 55 | 58 |
| Q3 | 70 | 64 |
| Maximum | 90 | 82 |
Compare the two distributions. (3 marks)
"The median mark for Class Q (58) is slightly higher than for Class P (55), which shows that on average, Class Q performed slightly better."
"The IQR for Class P is 70 - 40 = 30, while the IQR for Class Q is 64 - 42 = 22. Class Q has a smaller IQR, meaning the marks in Class Q were more consistent and less spread out."
"The range for Class P is 90 - 25 = 65, compared to 82 - 30 = 52 for Class Q, further confirming that Class P had greater variability in marks."
A histogram shows the lengths (l cm) of 80 leaves. The bar for the class 2 < l ≤ 4 has a frequency density of 5.
(a) How many leaves have lengths in the class 2 < l ≤ 4? (2 marks)
Class width = 4 - 2 = 2
Frequency = frequency density x class width = 5 x 2 = 10 leaves
(b) Another class is 4 < l ≤ 10 with a frequency of 24. What is the frequency density? (2 marks)
Class width = 10 - 4 = 6
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