Cambridge O Level Statistics: Range, interquartile range, variance and standard deviation (4040) – 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 are for Cambridge O Level Statistics (4040), aligned to the syllabus for examination in 2025, 2026 and 2027. They cover section 5.2: range, interquartile range, variance and standard deviation, from lists, ungrouped tables and grouped tables, and the comparison of spread. Quartiles of a list use the halves method and grouped quartiles use interpolation. The variance divides by n. Give answers to 3 significant figures unless told otherwise.

Learn the methods in the study guide first. The course page is Cambridge O Level Statistics.

Questions

1. The numbers of emails received by a manager on nine days were 7, 12, 5, 9, 14, 8, 11, 6, 10. Find the range and the interquartile range. [3]

2. Calculate the variance and the standard deviation of 2, 4, 5, 7, 8, 10. [4]

3. A librarian records the number of books each of 40 pupils borrowed in a term.

Books x	1	2	3	4	5	6
Pupils f	3	8	14	10	4	1

(a) Calculate the mean number of books. [2] (b) Calculate the variance and the standard deviation. [3] (c) Find the range and the interquartile range. [3]

4. The masses of 50 parcels are summarised below.

Mass m (kg)	0 to under 2	2 to under 4	4 to under 8	8 to under 12
Frequency	10	18	16	6

(a) Estimate the mean mass. [3] (b) Estimate the standard deviation of the masses. [3]

5. The scores of 100 players in a game are grouped as follows.

Score	40 to under 50	50 to under 60	60 to under 70	70 to under 90	90 to under 120
Frequency	5	15	30	35	15

(a) Use linear interpolation to estimate the lower and upper quartiles. [4] (b) Hence estimate the interquartile range. [1] (c) Estimate the range, and explain why this can only be an estimate. [2]

6. The numbers of customers at two food stalls over ten days were:

Stall S: 22, 25, 26, 28, 29, 30, 31, 33, 34, 42 Stall T: 18, 21, 26, 30, 31, 31, 33, 36, 38, 46

(a) Stall S has a mean of 30. Calculate its standard deviation. [4] (b) Stall T has a mean of 31, a standard deviation of 7.73 and an interquartile range of 10. Compare the two stalls, giving a comment on both the average and the spread. [3]

7. The ages of seven members of a club are 15, 17, 18, 20, 21, 22, 24.

(a) Find the range and the interquartile range. [2] (b) The age 24 is wrongly entered as 48. Find the new range and interquartile range. [2] (c) The standard deviation changes from 2.87 to 10.4 (3 s.f.). Explain which measure of spread is least affected by the data entry error, and why. [2]

8. For ten values, Σx = 120 and Σx² = 1650.

(a) Find the mean, the variance and the standard deviation. [3] (b) One value was recorded as 15 but should have been 11. Find the correct mean and standard deviation. [4]

9. The times, in minutes, taken by two groups of 50 people to complete a puzzle are shown.

Time t	10 to under 20	20 to under 30	30 to under 40	40 to under 50	50 to under 60
Group A	4	12	20	10	4
Group B	10	14	12	9	5

For Group A, an estimate of the mean is 34.6, of the standard deviation is 10.4 and of the interquartile range is 14.4.

(a) Estimate the mean and the standard deviation for Group B. [5] (b) Estimate the interquartile range for Group B. [3] (c) Compare the two groups, referring to both average and spread. [3]

10. Five numbers are 4, 7, 9, 12 and k. Their mean is 8.

(a) Find k. [2] (b) Find the variance and the standard deviation of the five numbers. [3]

11. A property agent describes the prices of houses in a village. Most cost between 150 and 200 thousand, but one costs 2 million.

(a) Which of the range and the interquartile range should the agent quote to describe the typical spread? Give a reason. [2] (b) The standard deviation of the delivery times to the village is 4.2 minutes. Write down the variance, with its unit. [1] (c) A data set has a standard deviation of 0. What does this tell you about the values? [1]

Answers

1. Ordered: 5, 6, 7, 8, 9, 10, 11, 12, 14. Range = 14 − 5 = 9 [1]. The median is 9, so the lower half is 5, 6, 7, 8 and the upper half is 10, 11, 12, 14. Q1 = 6.5 and Q3 = 11.5 [1]. IQR = 11.5 − 6.5 = 5 [1]. [3] Examiner insight: Sort the data first. Quartiles taken from an unsorted list will be wrong.

2. Mean = 36 ÷ 6 = 6 [1]. Squared deviations: 16, 4, 1, 1, 4, 16, which total 42 [1]. Variance = 42 ÷ 6 = 7 [1]. Standard deviation = √7 = 2.65 [1]. [4] Examiner insight: Show the squared deviations so that a slip can be seen. Do not stop at 7 when the question asks for the standard deviation.

3. (a) Σfx = 3 + 16 + 42 + 40 + 20 + 6 = 127 [1]. Mean = 127 ÷ 40 = 3.175 (3.18 to 3 s.f.) [1]. [2] (b) Σfx² = 3 + 32 + 126 + 160 + 100 + 36 = 457 [1]. Variance = 457/40 − 3.175² = 11.425 − 10.080625 = 1.34 [1]. Standard deviation = √1.344375 = 1.16 [1]. [3] (c) Range = 6 − 1 = 5 [1]. Cumulative frequencies are 3, 11, 25, 35, 39, 40. The 10th and 11th values are both 2, so Q1 = 2. The 30th and 31st values are both 4, so Q3 = 4 [1]. IQR = 4 − 2 = 2 [1]. [3] Examiner insight: Keep the unrounded mean in the variance formula: 3.18 in place of 3.175 changes the third significant figure.

4. (a) Midpoints 1, 3, 6, 10 [1]. Σfx = 10 + 54 + 96 + 60 = 220 [1]. Mean = 220 ÷ 50 = 4.4 kg [1]. [3] (b) Σfx² = 10 + 162 + 576 + 600 = 1348 [1]. Variance = 1348/50 − 4.4² = 26.96 − 19.36 = 7.6 [1]. Standard deviation = √7.6 = 2.76 kg [1]. [3] Examiner insight: Midpoints, not class boundaries, go into the table. State that the results are estimates, since the values inside each class are unknown.

5. (a) Cumulative frequencies: 5, 20, 50, 85, 100. The quartile positions are 25 and 75 [1]. Q1 is in the class 60 to under 70, with 20 before it: Q1 = 60 + (25 − 20)/30 × 10 = 61.7 [1]. Q3 is in the class 70 to under 90, with 50 before it [1]: Q3 = 70 + (75 − 50)/35 × 20 = 84.3 [1]. [4] (b) IQR = 84.3 − 61.7 = 22.6 [1]. [1] (c) Estimated range = 120 − 40 = 80 [1]. It is only an estimate because the actual smallest and largest scores are not known: they could be anywhere inside the first and last classes [1]. [2] Examiner insight: Write the position, the class and the fraction. A bare final figure is risky here.

6. (a) Deviations from 30: −8, −5, −4, −2, −1, 0, 1, 3, 4, 12 [1]. Squares: 64, 25, 16, 4, 1, 0, 1, 9, 16, 144 [1]. Sum = 280, so variance = 28 [1]. Standard deviation = √28 = 5.29 [1]. [4] (b) The means are similar (30 and 31), so on average the stalls serve about the same number of customers [1]. Stall T has the larger standard deviation (7.73 against 5.29) [1], so its daily customers are more variable and Stall S is more consistent. The comment must be about customers per day [1]. [3] Examiner insight: Name the measure, quote both values and use the context. “S is smaller” on its own says little.

7. (a) Range = 24 − 15 = 9 [1]. The median is 20, so Q1 = 17 (median of 15, 17, 18) and Q3 = 22 (median of 21, 22, 24). IQR = 22 − 17 = 5 [1]. [2] (b) The data is now 15, 17, 18, 20, 21, 22, 48. Range = 48 − 15 = 33 [1]. The quartiles stay at 17 and 22, so the IQR is still 5 [1]. [2] (c) The interquartile range [1]. The wrong value of 48 more than triples both the standard deviation and the range, but does not move the IQR, because the IQR ignores the top quarter of the data [1]. [2] Examiner insight: Give a reason that refers to this data and quote the figures that changed.

8. (a) Mean = 120 ÷ 10 = 12 [1]. Variance = 1650/10 − 12² = 165 − 144 = 21 [1]. Standard deviation = √21 = 4.58 [1]. [3] (b) Corrected Σx = 120 − 15 + 11 = 116 [1]. Corrected Σx² = 1650 − 225 + 121 = 1546 [1]. Mean = 11.6 [1]. Variance = 1546/10 − 11.6² = 154.6 − 134.56 = 20.04, so the standard deviation is √20.04 = 4.48 [1]. [4] Examiner insight: Correct both totals before you recompute. A frequent slip is to correct Σx but leave Σx² unchanged.

9. (a) Midpoints 15, 25, 35, 45, 55 [1]. Σfx = 150 + 350 + 420 + 405 + 275 = 1600 [1], so the mean is 1600 ÷ 50 = 32 [1]. Σfx² = 2250 + 8750 + 14700 + 18225 + 15125 = 59050 [1]. Variance = 59050/50 − 32² = 1181 − 1024 = 157, so the standard deviation = √157 = 12.5 [1]. [5] (b) Cumulative frequencies: 10, 24, 36, 45, 50. The positions are 12.5 and 37.5 [1]. Q1 = 20 + (12.5 − 10)/14 × 10 = 21.8 and Q3 = 40 + (37.5 − 36)/9 × 10 = 41.7 [1]. IQR = 41.7 − 21.8 = 19.9 [1]. [3] (c) Group A has the slightly higher mean (34.6 against 32), so its members were slower on average [1]. Group B has the larger standard deviation (12.5 against 10.4) [1] and the larger IQR (19.9 against 14.4), so its times are more spread out and Group A is more consistent [1]. [3] Examiner insight: Quote the spread measure for each group and say what it shows about times. A mean alone leaves half of a “compare” answer unfinished.

10. (a) The total is 5 × 8 = 40 [1]. So k = 40 − (4 + 7 + 9 + 12) = 40 − 32 = 8 [1]. [2] (b) Deviations: −4, −1, 1, 4, 0 [1]. Squares total 16 + 1 + 1 + 16 + 0 = 34, so the variance is 34 ÷ 5 = 6.8 [1]. Standard deviation = √6.8 = 2.61 [1]. [3] Examiner insight: Use your part (a) value in part (b) and show the deviations, so that each step can be checked.

11. (a) The interquartile range [1]. The house at 2 million is an extreme value that would make the range very large, while the IQR describes the middle half of the prices and is not affected by it [1]. [2] (b) Variance = 4.2² = 17.64 minutes² [1]. [1] (c) All the values are equal (there is no spread) [1]. [1] Examiner insight: The variance is in squared units, so include the unit in part (b). Name the extreme value in part (a).

Where marks are usually lost

• Not sorting the data before finding quartiles.
• Giving the variance when the question asks for the standard deviation, or the other way round.
• Using class boundaries instead of midpoints for grouped estimates.
• Comparing data sets without quoting values or using the context.

Next steps

• Review the revision notes and the study guide.
• Visit the course page and tick off the checklist.
• Try the free 10-minute diagnostics.
• Book a free trial class.

Official syllabus

Cambridge O Level Statistics 4040, syllabus for examination in 2025, 2026 and 2027, from Cambridge Assessment International Education, part of Cambridge University Press & Assessment. These questions cover section 5.2, measures of dispersion.