Cambridge IGCSE Statistics: Standard deviation and choosing a measure of dispersion (0479) – 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 outcomes 5.6 and 5.7 of Topic 5 in the Cambridge IGCSE Statistics (0479) syllabus for examination in 2027. Read the study guide first, or use the revision notes for recall. The course page is Cambridge IGCSE Statistics and the printable list is the syllabus checklist. Show your working and give answers to 3 significant figures unless told otherwise.

Questions

1. Five parcels weigh 20, 24, 25, 29 and 32 kg. Calculate the standard deviation of the weights. [3]

2. For eight values, Σx = 96 and Σx² = 1250. Calculate the standard deviation. [3]

3. The table shows the number of passengers in each of 40 cars that stopped at a checkpoint.

Passengers, x	1	2	3	4	5
Frequency, f	8	14	10	6	2

(a) Calculate the mean number of passengers. [2] (b) Calculate the standard deviation. [3]

4. The ages of 50 customers at a café are shown.

Age, x (years)	10 ≤ x < 20	20 ≤ x < 30	30 ≤ x < 40	40 ≤ x < 50	50 ≤ x < 60
Frequency	6	14	20	8	2

(a) Calculate an estimate of the mean age. [2] (b) Calculate an estimate of the standard deviation. [3] (c) Explain why your answers are estimates. [1]

5. Four numbers are 2, 6, 7 and x. Their mean is 6. Find x, and then calculate the standard deviation of the four numbers. [4]

6. A stall records the number of customers on seven days: 4, 6, 7, 8, 9, 11 and 35.

(a) Calculate the range and the interquartile range. [3] (b) Calculate the standard deviation. [3] (c) The stall owner wants one measure of how much the daily numbers usually vary. Choose a measure and justify your choice. [2]

7. A table of waiting times, in minutes, has a final class “60 or more”. Explain why the range and the standard deviation cannot be calculated from the table, and name one measure of dispersion that could still be estimated. [3]

8. Two bus routes have the same mean journey time of 18.0 minutes. Route A has a standard deviation of 2.1 minutes and Route B has a standard deviation of 5.6 minutes.

(a) Which route has the more consistent journey time? Give a reason. [2] (b) State one advantage of using the standard deviation rather than the range to compare the routes. [2]

9. A student finds the standard deviation of 2, 4, 7 and 11 using Σx = 24 and Σx² = 190. The student’s working is: 190/4 − 24/4 = 47.5 − 6 = 41.5, so the standard deviation is √41.5 = 6.44.

(a) Identify the error. [1] (b) Calculate the correct standard deviation. [3]

10. The number of repeat visits made by 25 customers is summarised below. The frequency k is unknown.

Visits, x	1	2	3	4	5
Frequency	4	9	k	6	3

The mean number of visits is 2.8.

(a) Find k. [3] (b) Calculate the standard deviation. [4] (c) A second survey also has a mean of 2.8 visits and a standard deviation of 1.9. Which survey shows the more consistent number of visits? [1]

11. (a) A data set has a standard deviation of zero. What can you say about its values? [1] (b) Explain why a standard deviation can never be negative. [1]

Answers

1. Mean = 130 / 5 = 26 [1]. Squared deviations: 36, 4, 1, 9, 36, which sum to 86 [1]. SD = √(86/5) = √17.2 = 4.15 kg [1] Examiner insight: Writing the sum Σ(x − x̄)² on its own line makes the method visible, which helps if a later slip occurs.

2. Mean = 96 / 8 = 12 [1]. Variance = 1250/8 − 12² = 156.25 − 144 = 12.25 [1]. SD = 3.5 [1] Examiner insight: The subtraction in the second formula is the mean of the squares minus the square of the mean. Subtracting 96/8 without squaring it is a frequent slip.

3. (a) Σfx = 8 + 28 + 30 + 24 + 10 = 100 [1]. Mean = 100 / 40 = 2.5 [1] (b) Σfx² = 8 + 56 + 90 + 96 + 50 = 300 [1]. Variance = 300/40 − 2.5² = 7.5 − 6.25 = 1.25 [1]. SD = √1.25 = 1.12 [1] Examiner insight: Build the fx and fx² columns in the table. A long single line of arithmetic is where frequency questions usually go wrong.

4. (a) Midpoints 15, 25, 35, 45, 55. Σfx = 90 + 350 + 700 + 360 + 110 = 1610 [1]. Mean = 1610 / 50 = 32.2 years [1] (b) Σfx² = 1350 + 8750 + 24 500 + 16 200 + 6050 = 56 850 [1]. Variance = 56850/50 − 32.2² = 1137 − 1036.84 = 100.16 [1]. SD = 10.0 years [1] (c) The exact ages are not known, so each person is treated as if they were at the midpoint of their class [1] Examiner insight: “Estimate” in the question means the midpoint method. Using the class boundary or the class width in place of the midpoint changes every later figure.

5. Sum of the four numbers = 4 × 6 = 24, so x = 24 − 15 = 9 [1]. Deviations: −4, 0, 1, 3, with squares 16, 0, 1, 9 summing to 26 [1]. Variance = 26/4 = 6.5 [1]. SD = √6.5 = 2.55 [1] Examiner insight: Use the unrounded value 6.5 under the root. State the value of x clearly before you use it in the next step.

6. (a) Range = 35 − 4 = 31 [1]. Lower quartile = 6 and upper quartile = 11 [1]. IQR = 11 − 6 = 5 [1] (b) Σx = 80 and Σx² = 16 + 36 + 49 + 64 + 81 + 121 + 1225 = 1592 [1]. Variance = 1592/7 − (80/7)² = 227.43 − 130.61 = 96.82 [1]. SD = 9.84 [1] (c) The IQR [1], because the day with 35 customers is far from the others and inflates the range and the standard deviation, whereas the IQR describes the usual spread of 5 [1] Examiner insight: A choice with no reason scores little. Quote the 35 and the figures 31, 9.84 and 5 to show why the IQR is better here.

7. The range needs the largest value, and the final class has no upper limit [1]. The standard deviation needs a midpoint for every class, and an open class has no midpoint [1]. The interquartile range could still be estimated, provided both quartiles lie in classes with limits [1] Examiner insight: Give both reasons separately. “The class is open” alone does not say what each measure needs.

8. (a) Route A [1], because its standard deviation is smaller, so its journey times are closer to the mean [1] (b) The standard deviation uses every journey time [1], whereas the range depends on only the longest and the shortest, so one unusual journey could change it a lot [1] Examiner insight: A comparison needs both figures (2.1 and 5.6) and a comparative word, not only the name of the route.

9. (a) The student subtracted Σx/n (6) instead of its square (6² = 36) [1] (b) Mean = 24/4 = 6 [1]. Variance = 47.5 − 36 = 11.5 [1]. SD = √11.5 = 3.39 [1] Examiner insight: Say exactly what was wrong. “Calculation error” is too vague to be credited.

10. (a) Σf = 22 + k and Σfx = 4 + 18 + 3k + 24 + 15 = 61 + 3k [1]. 61 + 3k = 2.8(22 + k) = 61.6 + 2.8k [1]. 0.2k = 0.6, so k = 3 [1] (b) Σfx² = 4 + 36 + 27 + 96 + 75 = 238 [1]. Σf = 25 and Σfx = 70 [1]. Variance = 238/25 − 2.8² = 9.52 − 7.84 = 1.68 [1]. SD = 1.30 [1] (c) The first survey [1], because its standard deviation (1.30) is smaller than 1.9. Examiner insight: Form an equation from the mean and solve it. Setting out the equation shows the method, whereas a bare value of k gives an examiner little to credit.

11. (a) All the values are the same [1] (b) The standard deviation is the square root of a mean of squared quantities, which cannot be negative [1] Examiner insight: For part (a), “no spread” is a restatement of the question. State the property of the values themselves.

Where marks are usually lost

• Giving the variance and not taking the square root.
• Squaring Σx instead of finding (Σx / n)², or subtracting the mean without squaring it.
• Dividing by the number of classes or table entries, not Σf.
• Using class boundaries or class widths instead of midpoints.
• Rounding the mean or variance early.
• Omitting the word “estimate” when working from a grouped table.
• Naming a measure of dispersion with no reason from the context.
• Judging consistency from the means, when the standard deviations are what differ.
• Writing only the final answer where the question asks you to show working.

Next steps

• Read the study guide again for the worked examples.
• Use the revision notes to memorise both formulae.
• Open the Cambridge IGCSE Statistics course page and the syllabus checklist.
• Try the free 10-minute diagnostics.
• Book a free trial class for help with this topic.

Official syllabus

Cambridge IGCSE Statistics (0479) syllabus for examination in 2027, Cambridge University Press and Assessment (Cambridge International Education), Topic 5, outcomes 5.6 and 5.7.