AQA A-Level Mathematics: Use of data in statistics (7357) – Practice Questions

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs – examination boards hold copyright in their own papers. Use these alongside the official past papers from your board or school.

These questions cover section 3.21: Use of data in statistics (including 3.21.1, Large data set) of the AQA A-level Mathematics (7357) specification, version 1.3 (31 January 2018), for A-level exams from June 2018 onwards. The specification does not place 3.21 under one paper, but the statistics sections it supports (K to O) are assessed on Paper 3. A calculator is required in every 7357 paper. All data here are invented; none is the AQA large data set. Questions 9 to 11 are extended problems.

Learn the methods first in the Use of data study guide and the Use of data revision notes. The 7357 course hub and the printable checklist show where this topic fits, and the free 10-minute diagnostics help you find weak spots.

Questions

1. State two things, other than its numerical values, that you should know about a large data set before analysing it. [2]

2. A subset of a data set gives these race times, in seconds:

52.4, 49.8, 55.1, 50.6, 53.9, 48.7, 51.2, 54.3

Use your calculator to find the mean and the standard deviation (divisor n). [3]

3. A spreadsheet summary of 50 values gives Σx = 3125 and Σx² = 199 362.5.

(a) Calculate the mean and the standard deviation (divisor n). [3] (b) The spreadsheet’s standard deviation function gives 9.09. Explain the difference. [1]

4. A data set in a spreadsheet has 800 rows.

(a) Describe how to use the spreadsheet to choose a simple random sample of 25 rows. [2] (b) Give one reason for using a spreadsheet, rather than a calculator, to explore the whole data set. [1]

5. Part of a cycling survey is shown.

ID	Age (years)	Distance cycled last week (km)
1	34	42.5
2	230	18.0
3	51
4	27	0
5	45	310

For each of rows 2, 3, 4 and 5, state whether an entry needs attention and what you would do with it. [4]

6. A bar chart shows annual visitors to a park, in thousands: 48.6 (2021), 50.2 (2022), 52.4 (2023) and 51.0 (2024). The vertical axis starts at 48.

(a) Calculate the percentage increase from 2021 to 2023. [2] (b) Explain, with figures, why the chart is misleading. [2] (c) Suggest one improvement to the chart. [1]

7. Daily screen time, in minutes, was recorded for two groups.

Group	n	Mean	Standard deviation	Median	IQR
P	60	204	61	185	70
Q	55	163	38	160	45

(a) What does the mean and median of Group P suggest about the shape of its distribution? [1] (b) State which measures you would use to compare the groups, and make two comparisons in context. [3]

8. Software reports that 24 values of weekly hours of paid work have mean 9.5 and Σx² = 2620. Three of the values are 0, but these were codes meaning “no response”.

(a) Calculate the mean of the valid values. [2] (b) Calculate the standard deviation (divisor n) of the valid values. [2] (c) Comment on the effect the codes had on the software’s results. [1]

9. A student has an invented data set of 1,200 households with floor area and weekly electricity use. She wants to know whether larger homes use more electricity.

(a) Write a precise question for the investigation and name the two variables. [2] (b) Describe how to take a random sample of 40 households using a spreadsheet, and say why a random sample is preferred. [2] (c) Name a suitable diagram and describe what it would show if larger homes use more. [2] (d) The diagram shows strong positive correlation. The student concludes that a bigger home causes higher use. Comment, and suggest a next step in the problem-solving cycle. [2]

10. The commuting distances, in km, of 12 employees chosen from a data set are:

3.2, 5.8, 7.4, 8.1, 9.6, 10.3, 11.7, 12.5, 14.0, 15.8, 18.6, 58.0

(a) Find the mean and the standard deviation (divisor n). [3] (b) An outlier is a value more than 2 standard deviations from the mean. Show that 58.0 is an outlier. [2] (c) Find the mean and the standard deviation with 58.0 removed. [2] (d) The 58.0 km value is genuine. Find the median and IQR of all 12 values, using the median of each half for the quartiles, and explain why these are better summaries here. [3]

11. An invented data set of 600 households has mean weekly electricity use 73.2 kWh. Two students each take a simple random sample of 20 households. Student A’s sample mean is 74.6 kWh. Student B’s sample gives Σx = 1418 and Σx² = 102 085.

(a) Find the mean and the standard deviation (divisor n) of Student B’s sample. [3] (b) Explain why the two sample means differ from each other and from 73.2. [2] (c) Student B says her sample shows that households in the data set use less than 73.2 kWh on average. Comment. [1] (d) Describe how a spreadsheet could show how much the mean of a sample of 20 varies. [2]

Answers

1. Any two, for example: what each variable measures and its units [1]; how missing values are recorded (or what one row represents, or how the data were collected) [1]. Examiner insight: “It is big” or “it is real” describes nothing specific; name a feature you would check.

2. x̄ = 416.0/8 = 52.0 s [1] σ² = 21 669/8 − 52.0² = 4.625 [1] σ = 2.15 s (3 s.f.) [1] Examiner insight: Write down x̄ and σ separately; a sentence with no values cannot show which calculator output you read.

3. (a) x̄ = 3125/50 = 62.5 [1] σ² = 199 362.5/50 − 62.5² = 3987.25 − 3906.25 = 81 [1] σ = 9 [1] (b) The software uses divisor n − 1: s = 9 × √(50/49) = 9.09 [1]. Examiner insight: Square the mean before subtracting; σ² = Σx²/n − x̄ is the slip that turns a correct start into a wrong answer.

4. (a) Add a column with =RAND() in every row and sort the data by that column [1]; take the first 25 rows [1]. (b) It handles all 800 rows quickly and can filter, chart and recalculate (any one) [1]. Examiner insight: “Pick randomly” is not a method; say how every row gets an equal chance.

5. Row 2: age 230 is impossible; correct it from the source or remove it [1]. Row 3: distance is missing; exclude it from distance calculations and do not record 0 [1]. Row 4: 0 km is plausible (the person did not cycle), so keep it unless 0 is a missing code [1]. Row 5: 310 km is extreme but possible; check it, and keep it if genuine [1]. Examiner insight: Each row needs a decision and a reason; “remove all odd values” gives no reasoning to credit.

6. (a) (52.4 − 48.6)/48.6 × 100 [1] = 7.82% (3 s.f.) [1] (b) The bars show only the part above 48: 0.6 for 2021 and 4.4 for 2023, so the 2023 bar is about 7.3 times as tall [1]. The true ratio is 52.4/48.6 = 1.08, an increase under 8% [1]. (c) Start the axis at 0, or show a clear break in the axis [1]. Examiner insight: “The axis is wrong” is too vague; say where it starts and what that does to the bar heights.

7. (a) Mean > median, so the distribution is positively skewed [1]. (b) Median and IQR, because P is skewed and medians resist extreme values [1]. Group P had a higher median screen time than Q (185 against 160 minutes) [1]. P’s screen times varied more (IQR 70 against 45 minutes) [1]. Examiner insight: Each comparison should name the group, the direction and the context; “P is higher” alone is incomplete.

8. (a) Total = 24 × 9.5 = 228, and the zeros add nothing [1]; x̄ = 228/21 = 10.9 hours (3 s.f.) [1] (b) σ² = 2620/21 − (228/21)² = 6.884… [1]; σ = 2.62 hours (3 s.f.) [1] (c) The codes lowered the mean and inflated the spread: the software’s σ was 4.35 hours [1]. Examiner insight: Reduce n to 21 in both Σx/n and Σx²/n; changing it in only one place gives a wrong σ.

9. (a) “In this data set, do households with larger floor areas use more electricity per week?” [1] Variables: floor area and weekly electricity use [1]. (b) Give each row a =RAND() value, sort by it and take the first 40 [1]. Random selection avoids bias, so the sample is more likely to represent the 1,200 households [1]. (c) A scatter diagram of electricity use against floor area [1]; points would trend upwards from left to right (positive correlation) [1]. (d) Correlation does not show causation; for example, larger homes may hold more people, who use more electricity [1]. Next step: investigate a third variable (such as household size), or repeat with a new sample [1]. Examiner insight: Generic steps such as “collect data” earn little; tie each step to the context.

10. (a) x̄ = 175/12 = 14.6 km (3 s.f.) [1] σ² = 4811.24/12 − (175/12)² = 188.26 [1]; σ = 13.7 km (3 s.f.) [1] (b) Upper limit = 14.58 + 2 × 13.72 = 42.0 [1]; 58.0 > 42.0, so 58.0 is an outlier [1]. (c) x̄ = 117/11 = 10.6 km [1]; σ = 4.29 km (3 s.f.) [1] (d) Median = (10.3 + 11.7)/2 = 11.0 km [1]; Q₁ = 7.75, Q₃ = 14.9, so IQR = 7.15 km [1]. The value is genuine, so it stays; the median and IQR are not distorted by it, while the mean and σ are [1]. Examiner insight: In (b), state the limit and the comparison; “58 is much bigger” is not a check.

11. (a) x̄ = 1418/20 = 70.9 kWh [1] σ² = 102 085/20 − 70.9² = 5104.25 − 5026.81 = 77.44 [1]; σ = 8.8 kWh [1] (b) Each sample contains different, randomly chosen households [1]. A sample mean only estimates the population mean, and it varies from sample to sample [1]. (c) The data set’s mean is 73.2 kWh; B’s lower mean reflects sampling variation, so her claim is wrong [1]. (d) Draw many samples of 20 (for example 100), recording each sample’s mean [1]; plot a histogram of the means, or find their range or standard deviation [1]. Examiner insight: State in context that a different sample gives a different result.

Where marks are usually lost

• Counting missing-value codes as 0 in a mean or standard deviation.
• Forgetting to reduce n after removing values.
• Writing σ² = Σx²/n − x̄ instead of subtracting x̄².
• Giving σ when s is asked for, or the reverse, without saying which.
• Removing a genuine extreme value just because it is large.
• Calling a chart misleading without saying where the axis starts.
• Comparing groups with no context or no measure of spread.
• Treating one sample’s result as a fact about the population.

Next steps

• Recap with the Use of data revision notes.
• Reread the Use of data study guide.
• Related: data presentation practice and sampling practice.
• The 7357 course hub and the printable checklist.
• Try all free 10-minute diagnostics.
• Book a free trial class.

Official syllabus

AQA A-level Mathematics (7357) specification, version 1.3 (31 January 2018), for A-level exams June 2018 onwards, published by AQA. Section 3.21, Use of data in statistics, including 3.21.1, Large data set.