AQA A-Level Mathematics: L: Data presentation and interpretation (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 L: Data presentation and interpretation (L1 to L4) of the AQA A-level Mathematics (7357) specification, version 1.3, for A-level exams from June 2018 onwards. The specification lists Section L under Paper 3. A calculator is required in every 7357 paper, so every question here assumes you have one with statistics functions. Standard deviations use the divisor n unless a question says otherwise.

Learn the methods first in the Data presentation study guide and the Data presentation revision notes. The 7357 course hub and the printable checklist list the rest of the course, and the free 10-minute diagnostics show which topic to practise next.

Questions

1. In a grouped frequency table, the class 20 ≤ m < 35 has frequency 27.

(a) Calculate the frequency density of this class. [1] (b) On the histogram, the bar for 35 ≤ m < 40 has frequency density 3.6. Find the frequency of that class. [1]

2. On a histogram, the bar for the class 40 ≤ x < 50 is 2 cm wide and 6 cm tall. It represents 24 people. Another bar is 3 cm wide and 2.5 cm tall. Find the number of people it represents. [3]

3. For a sample of 25 values, Σx = 1150 and Σx² = 53 800. Calculate the mean and the standard deviation. [3]

4. Across a group of towns there is strong positive correlation between the number of cafés and the number of bookshops. A councillor says that opening more cafés will cause more bookshops to open. Comment on this claim. [2]

5. The commuting times, in minutes, of two groups of workers are summarised below.

	Minimum	Q₁	Median	Q₃	Maximum
Group A	12	20	26	35	58
Group B	15	24	28	31	40

(a) Compare the commuting times of the two groups, making two comparisons in context. [2] (b) An outlier is defined as a value more than 1.5 × IQR above Q₃. Show that the maximum of Group A is an outlier. [2]

6. The reaction times, in milliseconds, of 10 people are:

245, 262, 238, 251, 270, 244, 259, 248, 330, 255

An outlier is defined as a value more than 2 standard deviations from the mean. Use your calculator to find the mean and standard deviation, and identify any outliers. [4]

7. The lengths of 60 phone calls are shown.

Length t (min)	0 ≤ t < 2	2 ≤ t < 5	5 ≤ t < 10	10 ≤ t < 20
Frequency	14	22	18	6

Estimate the mean and standard deviation of the call lengths. [4]

8. On 20 days a café records the maximum temperature, T °C, and the number of cold drinks sold, D. The temperatures range from 12 °C to 28 °C. The regression line of D on T is D = 4.2T − 18.

(a) Interpret the value 4.2 in context. [1] (b) Explain why the line should not be used to predict sales on a day with a maximum temperature of 2 °C. [2] (c) The data were collected from two branches, one on a beach and one in a city centre. The scatter diagram shows two separate clusters of points. Explain why a single regression line may be misleading here. [2]

9. A researcher has 30 values with Σx = 1512 and Σx² = 79 272.

(a) Calculate the mean and standard deviation. [3] (b) While cleaning the data, the researcher finds that one value was typed as 95 instead of 59. Calculate the corrected mean and standard deviation. [3]

10. Ages of visitors to a museum are grouped into classes of widths 5, 10, 20 and 40 years. A student draws a bar chart with frequency on the vertical axis and bars of matching widths.

(a) Explain why this diagram is misleading and state what should be plotted instead. [2] (b) Suggest a suitable diagram for comparing the ages of visitors in two different years. [1]

11. The masses of 80 parcels are shown.

Mass m (kg)	0 ≤ m < 1	1 ≤ m < 2	2 ≤ m < 4	4 ≤ m < 7	7 ≤ m < 10
Frequency	12	20	28	15	5

(a) Calculate the frequency density of each class. [2] (b) Estimate the number of parcels with masses between 1.5 kg and 5 kg. [2] (c) A parcel is chosen at random from these 80. Find the probability that its mass is at least 4 kg. [1] (d) Estimate the mean and standard deviation of the masses. [3] (e) Use linear interpolation to estimate the median mass. [2]

12. A class of 15 students had a test mean of 62 and standard deviation of 5. A second class of 10 students had a mean of 70 and standard deviation of 4.

(a) Find the mean of all 25 scores. [2] (b) Find the standard deviation of all 25 scores. [4] (c) Explain why the combined standard deviation is larger than that of either class. [1] (d) A histogram of all 25 scores is drawn. Suggest what feature it might show. [1]

Answers

1. (a) 27 ÷ 15 = 1.8 [1] (b) 3.6 × 5 = 18 [1]. [2] Examiner insight: write the class width you used; a wrong width with no working leaves nothing to credit.

2. Area of the first bar = 2 × 6 = 12 cm² for 24 people [1], so 1 cm² represents 2 people [1]. Second bar: 3 × 2.5 = 7.5 cm², so 7.5 × 2 = 15 people [1]. [3] Examiner insight: using bar heights alone (6 for 24, so 2.5 for 10) ignores width and gives a wrong answer.

3. x̄ = 1150 ÷ 25 = 46 [1]. σ² = 53 800/25 − 46² = 2152 − 2116 = 36 [1], so σ = 6 [1]. [3] Examiner insight: show Σx²/n − x̄² before the square root; a bare σ with an arithmetic slip earns no method credit.

4. Correlation does not imply causation, so the claim is not supported by the data [1]. A third variable, such as the population of the town, could increase both the number of cafés and the number of bookshops [1]. [2] Examiner insight: naming a plausible third variable in context is stronger than repeating “correlation is not causation”.

5. (a) The median for Group B (28 minutes) is higher than for Group A (26 minutes), so Group B’s commutes are slightly longer on average [1]. Group A’s IQR (15 minutes) is larger than Group B’s (7 minutes), so Group A’s commuting times are more varied [1]. (b) IQR for A = 35 − 20 = 15, so the limit is 35 + 1.5 × 15 = 57.5 [1]. 58 > 57.5, so 58 is an outlier [1]. [4] Examiner insight: each comparison needs the numbers and a sentence about commuting; “B is higher” alone is not an interpretation.

6. Mean = 2602 ÷ 10 = 260.2 ms [1]. σ = 24.9 ms (3 s.f.) [1]. Limits: 260.2 ± 2 × 24.94, giving 210.3 and 310.1 [1]. 330 > 310.1, so 330 ms is an outlier; no value is below 210.3 [1]. [4] Examiner insight: keep the unrounded σ for the limits; state both limits so it is clear you checked for low outliers too.

7. Midpoints 1, 3.5, 7.5, 15; Σf = 60, Σfx = 316, Σfx² = 2646 [1]. x̄ = 316 ÷ 60 = 5.27 minutes (3 s.f.) [1]. σ² = 2646/60 − (316/60)² = 16.36… [1], so σ = 4.05 minutes (3 s.f.) [1]. [4] Examiner insight: using class upper boundaries instead of midpoints gives the wrong sums and loses the accuracy marks.

8. (a) For each 1 °C rise in maximum temperature, about 4.2 more cold drinks are sold [1]. (b) 2 °C is outside the observed range of 12 °C to 28 °C [1]. This is extrapolation, so the linear relationship may not hold there and the prediction is unreliable [1]. (c) The two branches form distinct sections of the population with different sales patterns [1]. One line through both clusters may show a correlation that does not hold within either branch, so each branch should be analysed separately [1]. [5] Examiner insight: in (b) quote the data range; “it is too cold” without reference to extrapolation misses the key point.

9. (a) x̄ = 1512 ÷ 30 = 50.4 [1]. σ² = 79 272/30 − 50.4² = 2642.4 − 2540.16 = 102.24 [1], so σ = 10.1 (3 s.f.) [1]. (b) New Σx = 1512 − 95 + 59 = 1476 and new Σx² = 79 272 − 95² + 59² = 73 728 [1]. x̄ = 1476 ÷ 30 = 49.2 [1]. σ = √(73 728/30 − 49.2²) = √36.96 = 6.08 (3 s.f.) [1]. [6] Examiner insight: you must update Σx² by subtracting 95² and adding 59²; adjusting only Σx leaves the standard deviation wrong.

10. (a) With unequal class widths, the area of each bar should represent frequency, but here wide classes look larger than they are [1]. Plot frequency density (frequency ÷ class width) on the vertical axis to draw a histogram [1]. (b) Box plots for the two years, drawn on the same scale [1]. [3] Examiner insight: “use a histogram” alone is not enough in (a); say what goes on the vertical axis.

11. (a) Frequency densities: 12, 20, 14, 5 and 1.67 (3 s.f.) [1] for all five correct, [1] for using frequency ÷ class width. (b) 0.5 × 20 + 28 + 1 × 5 [1] = 43 parcels [1]. (c) (15 + 5) ÷ 80 = 0.25 [1]. (d) Midpoints 0.5, 1.5, 3, 5.5, 8.5: Σfx = 245 and Σfx² = 1115 [1]. x̄ = 245 ÷ 80 = 3.06 kg (3 s.f.) [1]. σ = √(1115/80 − 3.0625²) = 2.14 kg (3 s.f.) [1]. (e) The median is the 40th value. Cumulative frequencies 12, 32, 60, so it lies in 2 ≤ m < 4 [1]. Median ≈ 2 + (8/28) × 2 = 2.57 kg (3 s.f.) [1]. [10] Examiner insight: in (b) only part of the 1–2 and 4–7 classes counts; adding whole classes is a common way to lose both marks.

12. (a) Totals: 15 × 62 = 930 and 10 × 70 = 700 [1]. Mean = 1630 ÷ 25 = 65.2 [1]. (b) For class 1, Σx² = 15(5² + 62²) = 58 035 [1]. For class 2, Σx² = 10(4² + 70²) = 49 160 [1]. Combined σ² = 107 195/25 − 65.2² = 36.76 [1], so σ = 6.06 (3 s.f.) [1]. (c) The class means differ by 8 marks, so the combined scores include the gap between the two groups as well as the spread within each one [1]. (d) Two peaks (two separate clusters), one for each class, because the data come from two distinct sections [1]. [8] Examiner insight: rebuild Σx² from σ² + x̄² for each group; averaging the two standard deviations is a method error.

Where marks are usually lost

• Bar heights treated as frequencies on histograms with unequal class widths.
• The area scale not found before reading a frequency from a histogram drawn to a scale.
• Whole classes counted when a question asks about part of a class.
• Σx² not updated when a value is corrected, added or removed.
• Combined standard deviations found by averaging the separate ones.
• Outlier limits not stated, so the conclusion is unsupported.
• Comparisons given without context, or with two location measures and no spread measure.
• Correlation described as causation.
• Predictions trusted outside the range of the data.
• Rounded values used in later steps, which shifts the final answer.

Next steps

• Data presentation revision notes
• Data presentation study guide
• Statistical sampling practice questions, for where your data come from
• AQA A-level Mathematics course hub
• Printable 7357 checklist
• 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 3.13, L: Data presentation and interpretation.