IB DP Mathematics: Analysis and Approaches – Sampling, data presentation, summary statistics and regression 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 – the IB holds copyright in its own papers. Use these alongside the official past papers available through your school or the IB store.

This practice set covers the descriptive statistics unit of IB Diploma Programme Mathematics: Analysis and Approaches. It is aligned to the IB Mathematics: analysis and approaches guide, first assessment 2021, syllabus sections 4.1–4.4 and 4.10, and every question is common to SL and HL. It follows the IB guide for first assessment 2021, which remains the examined syllabus until the new course is first assessed in May 2029, so it applies to the May and November 2026, 2027 and 2028 SL and HL sessions.

Paper 1 allows no technology; Paper 2 (and HL Paper 3) requires it. Questions are labelled “(calculator-free)” or “(calculator allowed)”. Give answers exactly or to 3 significant figures, and treat each data set as the population.

Learn the methods first in the statistics study guide and the revision notes. The IB DP Maths AA course hub and the printable syllabus checklist show where this unit sits.

Questions

1. (calculator-free) A café owner collects customer data.

(a) State whether each variable is discrete or continuous: (i) the number of items a customer buys; (ii) the time a customer waits to be served. [2] (b) She questions the first 30 customers to arrive on Monday morning. Name this sampling method. [1]

2. (calculator-free) A sports club has 540 members: 300 adults, 180 juniors and 60 seniors. A stratified sample of 36 members is chosen. Find the number of members chosen from each group. [3]

3. (calculator-free) For a set of reaction times, in milliseconds, the minimum is 15, Q₁ = 42, Q₃ = 58 and the maximum is 95. Determine whether the minimum and the maximum are outliers. [4]

4. (calculator-free) The prices of the items in a shop have mean £40 and standard deviation £6.

(a) Every price is increased by £3. Write down the new mean and the new standard deviation. [2] (b) Instead, every original price is increased by 10%. Find the new mean and the new standard deviation. [2]

5. (calculator-free) Twelve students recorded the number of books they read in a term:

1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 16

(a) Find the median. [1] (b) Find the lower quartile and the upper quartile. [2] (c) Show that 16 is an outlier. [2] (d) State the values at the ends of the whiskers on a box and whisker diagram of these data. [1]

6. (calculator allowed) The lengths, L cm, of 80 fish are shown.

L	10 ≤ L < 15	15 ≤ L < 20	20 ≤ L < 25	25 ≤ L < 30	30 ≤ L < 35
Frequency	8	18	26	20	8

(a) Write down the modal class. [1] (b) Estimate the mean length. [2] (c) Write down the cumulative frequencies, and hence estimate the median length using linear interpolation. [3] (d) Find the 90th percentile. [1]

7. (calculator allowed) The number of phone calls, x, received by a small office on each of 35 days is shown.

x	0	1	2	3	4	5
Frequency	3	7	10	8	5	2

Find the mean, the standard deviation and the variance of x. [4]

8. (calculator-free) On days with maximum temperature x °C, from 10 °C to 40 °C, a kiosk’s cold drink sales, y, have regression line of y on x, y = 1.8x + 12. The mean of x is 25.

(a) Find the mean number of drinks sold. [2] (b) Interpret the value 1.8 in context. [1] (c) Predict the number of drinks sold on a day with maximum temperature 30 °C. [1] (d) Explain why the line should not be used when x = −5. [1]

9. (calculator allowed) The age, x years, and price, y thousand pounds, of eight used cars of the same model are shown.

x	1	2	3	4	5	6	8	9
y	24.5	21.0	21.5	17.0	17.5	14.0	12.5	9.0

(a) Find Pearson’s product-moment correlation coefficient, r. [2] (b) Describe the correlation between age and price. [1] (c) Find the equation of the regression line of y on x. [2] (d) Use your line to estimate the price of a car that is 7 years old. [2] (e) A car of this model is priced at £14 000. Using an appropriate regression line, estimate its age. [2]

10. (calculator-free) Test scores for two classes are summarised below.

	Minimum	Q₁	Median	Q₃	Maximum
Class A	32	48	56	64	79
Class B	20	45	61	67	71

(a) Find the interquartile range for each class. [2] (b) Compare the two distributions. [2] (c) State which class’s scores could be normally distributed. Give a reason. [1] (d) Determine whether the score of 20 in Class B is an outlier. [1]

11. (calculator-free) On 10 days, a town records ice cream sales and the number of people treated for sunburn. For these data r = 0.87, and the critical value of r for this sample size is 0.632.

(a) State, with a reason, whether there is significant positive linear correlation. [1] (b) A newspaper claims that eating ice cream causes sunburn. Comment on this claim. [2] (c) Another data set has r = 0.05, yet its points lie close to a U-shaped curve. Explain how. [1]

12. (calculator allowed) Eight students’ marks in a test out of 40 are

14, 17, 18, 21, k, 25, 26, 30

where 21 ≤ k ≤ 25. The mean mark is 22.

(a) Find k. [2] (b) Find the median and the interquartile range. [2] (c) Find the standard deviation. [1] (d) The teacher converts each mark m to a scaled score using p = 2.5m − 5. Find the mean, the standard deviation and the interquartile range of p. [3] (e) A ninth student sits the test, and the mean mark of all nine is 23. Find this student’s mark. [2]

Answers

1. (a) (i) Discrete [1] (ii) Continuous [1] (b) Convenience sampling [1] Examiner insight: Naming two methods (“convenience or quota”) is a contradictory answer and loses the mark.

2. Sampling fraction = 36/540 = 1/15 [1]. Adults: 300/15 = 20, juniors: 180/15 = 12 [1], seniors: 60/15 = 4 [1] Examiner insight: Check that 20 + 12 + 4 = 36 before moving on.

3. IQR = 58 − 42 = 16 [1]; 1.5 × 16 = 24 [1]. Lower fence 42 − 24 = 18 and upper fence 58 + 24 = 82 [1]. Since 15 < 18 and 95 > 82, both the minimum and the maximum are outliers [1] Examiner insight: Fences from the median (50 ± 24) lose the fence mark and the conclusion that depends on it.

4. (a) Mean = £43 [1]; standard deviation = £6 [1] (b) Multiply by 1.1: mean = 1.1 × 40 = £44 [1]; standard deviation = 1.1 × 6 = £6.60 [1] Examiner insight: A new standard deviation of 9 in (a) loses that mark even with the mean correct.

5. (a) Median = (4 + 5)/2 = 4.5 [1] (b) Q₁ = (2 + 3)/2 = 2.5 [1]; Q₃ = (6 + 7)/2 = 6.5 [1] (c) IQR = 4, so the upper fence is 6.5 + 1.5 × 4 [1] = 12.5, and 16 > 12.5 [1] (d) 1 and 8 [1] Examiner insight: “Show that” needs the fence written down and compared with 16; “16 is much bigger” earns nothing.

6. (a) 20 ≤ L < 25 [1] (b) Σfx = 12.5(8) + 17.5(18) + 22.5(26) + 27.5(20) + 32.5(8) = 1810 [1]; mean ≈ 1810/80 = 22.6 cm [1] (c) Cumulative frequencies 8, 26, 52, 72, 80 [1]. The median is the 40th value, in 20 ≤ L < 25 [1]. Median ≈ 20 + (40 − 26)/26 × 5 ≈ 22.7 cm [1] (d) 90% of 80 = 72, reached at L = 30, so 30 cm [1] Examiner insight: Mid-interval values earn the method mark in (b); using class boundaries loses both marks.

7. Enter x with the frequencies as a second list [1]. Mean = 81/35 ≈ 2.31 [1]; standard deviation σ ≈ 1.33 [1]; variance ≈ 1.76 [1] Examiner insight: Squaring the rounded standard deviation gives 1.77, which is wrong in the third figure; square the unrounded value.

8. (a) The line passes through the mean point, so ȳ = 1.8 × 25 + 12 [1] = 57 [1] (b) For each 1 °C rise in maximum temperature, about 1.8 more drinks are sold on average [1] (c) y = 1.8 × 30 + 12 = 66 [1] (d) x = −5 is outside the data range (10 to 40), so this is unreliable extrapolation [1] Examiner insight: In (b), “the gradient is 1.8” is not an interpretation; the mark needs a rate of change in context.

9. (a) Using the GDC linear regression [1], r ≈ −0.977 [1] (b) Strong negative linear correlation [1] (c) a ≈ −1.79 and b ≈ 25.6 [1], so y = −1.79x + 25.6 [1] (d) y ≈ −1.788 × 7 + 25.62 [1] ≈ 13.1, so about £13 100 [1] (e) Use the regression line of x on y: x ≈ −0.534y + 13.9 [1]. With y = 14, x ≈ 6.42 years [1] Examiner insight: Rearranging the y on x line in (e) gives 6.50; it is the wrong method and earns neither mark.

10. (a) Class A: 64 − 48 = 16 [1]; Class B: 67 − 45 = 22 [1] (b) Class B has the higher median (61 > 56), so scored higher on average [1]. Class A has the smaller IQR, so its scores are more consistent [1] (c) Class A: median central in the box, whiskers almost equal (16 and 15), so roughly symmetric [1] (d) Lower fence = 45 − 1.5 × 22 = 12; 20 > 12, so 20 is not an outlier [1] Examiner insight: Each comparison mark needs a statement about both classes in context; listing two medians without comparing them earns nothing.

11. (a) Yes, because 0.87 > 0.632 [1] (b) Correlation does not imply causation [1]. A third variable, sunny weather, likely increases both [1] (c) r measures only linear correlation, so a strong curved relationship can still give r close to 0 [1] Examiner insight: In (a) the reason is the comparison 0.87 > 0.632; “yes” alone earns nothing.

12. (a) Σx = 8 × 22 = 176 [1], so k = 176 − 151 = 25 [1] (b) Ordered: 14, 17, 18, 21, 25, 25, 26, 30. Median = (21 + 25)/2 = 23 [1]; Q₁ = 17.5, Q₃ = 25.5, IQR = 8 [1] (c) σ ≈ 5.05 [1] (d) Mean = 2.5 × 22 − 5 = 50 [1]; standard deviation = 2.5 × 5.05 ≈ 12.6 [1]; IQR = 2.5 × 8 = 20 [1] (e) Total of nine marks = 9 × 23 = 207 [1], so the mark is 207 − 176 = 31 [1] Examiner insight: Subtracting 5 from the standard deviation or IQR in (d) loses those marks; a wrong σ from (c) can still earn follow-through for 2.5σ.

Where marks are usually lost

• Rearranging the y on x regression line to predict x from y.
• Measuring the outlier fence from the median rather than from the nearest quartile.
• Using class boundaries instead of mid-interval values for a grouped mean.
• Applying a constant change to the standard deviation or IQR, or scaling variance by k instead of k².
• Comparing distributions without context, or without comments on both centre and spread.
• Predicting outside the data range without saying it is extrapolation.

Next steps

• Recap with the statistics revision notes.
• Rework weak areas with the statistics study guide.
• See every unit in the IB DP Maths AA course hub and tick off topics on the printable syllabus checklist.
• Try all free 10-minute diagnostics.
• Book a free trial class.

Official syllabus

International Baccalaureate Organization, Diploma Programme, Mathematics: analysis and approaches guide, first assessment 2021 (published February 2019, updated November 2020).