IB DP Mathematics: Applications and Interpretation – Sampling, data presentation and summary statistics 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.

These practice questions cover sampling, data presentation and summary statistics for IB Diploma Programme Mathematics: Applications and Interpretation. They are aligned to the IB Mathematics: applications and interpretation guide, first assessment 2021, syllabus sections 4.1, 4.2 and 4.3, which are common content for SL and HL. They follow the IB guide for first assessment 2021, which remains the examined syllabus until the new course is first assessed in May 2029, so they apply to the May and November 2026, 2027 and 2028 sessions.

The guide lists every paper in this course as “technology required”, so every question is labelled (calculator allowed). Treat each data set as the population unless told otherwise, and give answers exactly or to 3 s.f.

Links: course hub · printable checklist · study guide · revision notes · statistics and probability overview

Questions

1. (calculator allowed) State whether each variable is discrete or continuous.

(a) The number of emails a person receives in a day. [1] (b) The mass of a parcel. [1] (c) The time taken to finish a crossword. [1]

2. (calculator allowed) Name the sampling method in each case.

(a) A shop picks a random number from 1 to 20, then surveys every 20th customer after that. [1] (b) A researcher interviews the first 30 people who walk into a cinema. [1] (c) An interviewer is told to question 25 men and 25 women and chooses whoever is passing. [1]

3. (calculator allowed) A company has 240 staff in production, 96 in sales and 64 in administration. A stratified sample of 50 staff is taken.

(a) Find the number of staff sampled from each department. [3] (b) State one advantage of this method over a simple random sample of 50 staff. [1]

4. (calculator allowed) The times, in seconds, that 40 people took to solve a puzzle have minimum 12, Q₁ = 42, median 49, Q₃ = 58 and maximum 95.

(a) Show that the minimum and the maximum are both outliers. [4] (b) The 12-second time came from a person who skipped half the puzzle. State, with a reason, whether to keep it. [1]

5. (calculator allowed) The number of books, x, read last month by 40 people is shown.

x	0	1	2	3	4
Frequency	4	9	13	9	5

(a) Find the mean. [2] (b) Find the median. [1] (c) Write down the mode. [1]

6. (calculator allowed) The daily maximum temperatures in a town over 30 days have mean 14.2 °C and standard deviation 3.5 °C. Each temperature is converted to degrees Fahrenheit using F = 1.8C + 32.

(a) Find the mean in °F. [2] (b) Find the standard deviation in °F. [1] (c) Find the variance of the temperatures in °F. [1]

7. (calculator allowed) The masses, m kg, of 80 parcels are shown.

Mass	0 ≤ m < 2	2 ≤ m < 4	4 ≤ m < 6	6 ≤ m < 8	8 ≤ m < 10
Frequency	9	23	27	15	6

(a) Write down the modal class. [1] (b) Find an estimate of the mean mass. [2] (c) Find an estimate of the standard deviation. [1] (d) Explain why your answer to (b) is only an estimate. [1]

8. (calculator allowed) The heights, h cm, of 120 seedlings are summarised below. The shortest seedling is 10 cm.

Height	h < 15	h < 20	h < 25	h < 30	h < 35	h < 40
Cumulative frequency	10	30	60	100	112	120

Use a cumulative frequency graph that joins (10, 0), (15, 10), (20, 30), … with straight lines.

(a) Write down the number of seedlings with 25 ≤ h < 30. [1] (b) Find the median height. [2] (c) Find Q₁, Q₃ and the interquartile range. [3] (d) Find the 80th percentile. [2] (e) Find the number of seedlings taller than 27 cm. [2] (f) A second tray of seedlings has median 23 cm and IQR 12 cm. Compare the two trays. [2]

9. (calculator allowed) The battery lives, in hours, of 15 phones are:

31, 34, 36, 37, 38, 40, 41, 41, 42, 43, 44, 45, 47, 51, 62

(a) Find the median, Q₁ and Q₃. [3] (b) Show that 62 is an outlier. [2] (c) A box and whisker diagram is drawn. Write down the values at the ends of the whiskers and state how 62 is shown. [2] (d) Comment on whether the battery lives may be normally distributed. [1]

10. (calculator allowed) A council wants the weekly exercise hours of a town’s residents. It posts a survey on a running club’s website and gets 200 responses. Twelve leave the hours blank; one says 400 hours.

(a) Explain why the sample is likely to be biased. [1] (b) State how the 12 blank responses should be handled when finding the mean. [1] (c) Explain why the 400-hour response is an error and state what should be done with it. [2]

11. (calculator allowed) Machine A fills bottles. The volumes, in ml, of 10 bottles from machine A are:

498, 502, 505, 499, 501, 503, 497, 500, 504, 501

For 15 bottles from machine B, the mean is 500.2 ml and the standard deviation is 4.6 ml.

(a) Find the mean and standard deviation for machine A. [2] (b) Compare the two machines. [2] (c) Machine A is adjusted so each of these 10 volumes falls by exactly 1 ml. Write down the new mean and standard deviation. [2] (d) Using the original data, find the mean volume of all 25 bottles, correct to one decimal place. [2] (e) An inspector checks every 25th bottle from the line after a random start. Name this method and give one weakness. [2]

Answers

1. (a) Discrete: it is a count. [1] (b) Continuous: it is measured. [1] (c) Continuous: time is measured. [1] Examiner insight: If a reason is asked for, “because it is a number” earns nothing; say “counted” or “measured”.

2. (a) Systematic sampling. [1] (b) Convenience sampling. [1] (c) Quota sampling. [1] Examiner insight: Part (c) is not stratified: the group sizes are fixed but people are not chosen at random.

3. (a) Total = 400, fraction sampled = 50/400 = 1/8 [1] Production 240 × 1/8 = 30 [1] Sales 12, administration 8 [1] (b) Each department is represented in proportion to its size, which a simple random sample does not guarantee. [1] Examiner insight: Show the fraction 50/400; if a bare answer is wrong, there is no method mark to fall back on.

4. (a) IQR = 58 − 42 = 16 [1] 1.5 × 16 = 24 [1] Lower boundary 42 − 24 = 18, and 12 < 18, so 12 is an outlier [1] Upper boundary 58 + 24 = 82, and 95 > 82, so 95 is an outlier [1] (b) Remove it: it is not a genuine time for the whole puzzle, so it is an error, not valid data. [1] Examiner insight: In a “show that”, write each boundary as a number and compare; “12 and 95 are far from the quartiles” scores zero.

5. (a) Σfx = 0 + 9 + 26 + 27 + 20 = 82 [1] x̄ = 82/40 = 2.05 books [1] (b) Cumulative frequencies 4, 13, 26, …; the 20th and 21st values are both 2, so median = 2 [1] (c) Mode = 2 [1] Examiner insight: Dividing 82 by 5 (the number of columns) instead of 40 keeps the Σfx mark but loses the answer mark; divide by Σf.

6. (a) 1.8 × 14.2 + 32 [1] = 57.56 = 57.6 °F (3 s.f.) [1] (b) 1.8 × 3.5 = 6.3 °F; adding 32 does not change the spread [1] (c) 6.3² = 39.69 = 39.7 (3 s.f.) [1] Examiner insight: Writing 1.8 × 3.5 + 32 = 38.3 loses (b), but squaring it correctly in (c) can still earn follow-through.

7. (a) 4 ≤ m < 6 [1] (b) Mid-interval values 1, 3, 5, 7, 9; Σfm = 9 + 69 + 135 + 105 + 54 = 372 [1] x̄ ≈ 372/80 = 4.65 kg [1] (c) GDC with mid-interval values and frequencies: σ ≈ 2.19 kg [1] (d) The actual masses are unknown; each is replaced by its class’s mid-interval value. [1] Examiner insight: Writing the modal class as “27” gives the frequency, not the class, and earns no mark.

8. (a) 100 − 60 = 40 [1] (b) 120/2 = 60th value [1] Median = 25 cm [1] (c) Q₁ at the 30th value = 20 cm [1] Q₃ at the 90th value: 25 + (90 − 60)/40 × 5 = 28.75 cm [1] IQR = 28.75 − 20 = 8.75 cm [1] (d) 0.8 × 120 = 96th value [1] 25 + (96 − 60)/40 × 5 = 29.5 cm [1] (e) Cumulative frequency at 27: 60 + (2/5) × 40 = 76 [1] 120 − 76 = 44 seedlings [1] (f) The first tray is taller on average (median 25 > 23 cm). [1] Its heights are less spread out (IQR 8.75 < 12 cm). [1] Examiner insight: In (e) the graph gives the number shorter than 27 cm; not subtracting from 120 loses the accuracy mark.

9. (a) Median = 8th value = 41 [1] Q₁ = median of the first seven values = 37 [1] Q₃ = median of the last seven values = 45 [1] (b) IQR = 8, upper boundary = 45 + 1.5 × 8 = 57 [1] 62 > 57, so 62 is an outlier [1] (c) Whiskers end at 31 and 51 [1] 62 is marked with a cross [1] (d) Box and whiskers are symmetric about the median (4 and 4; 10 and 10), so the data may be normally distributed. [1] Examiner insight: A whisker drawn to 62 loses the mark in (c) even after (b) shows it is an outlier.

10. (a) Running club website visitors probably exercise more than typical residents, so they are over-represented. [1] (b) Leave them out, find the mean from the other responses and say so. [1] (c) A week has only 7 × 24 = 168 hours, so 400 is impossible. [1] Remove it (or check it with the respondent) before finding the mean. [1] Examiner insight: “The sample is small” does not answer (a); give a reason why one group is more likely to respond.

11. (a) Mean = 501 ml [1] Standard deviation = 2.45 ml (3 s.f.) [1] (b) Machine A has a slightly higher mean volume (501 > 500.2 ml). [1] Machine A is more consistent (standard deviation 2.45 < 4.6 ml). [1] (c) Mean = 501 − 1 = 500 ml [1] Standard deviation = 2.45 ml (unchanged) [1] (d) Total for A = 10 × 501 = 5010 [1] (5010 + 15 × 500.2)/25 = 12513/25 = 500.5 ml [1] (e) Systematic sampling [1] A fault that repeats every 25 bottles could bias the sample. [1] Examiner insight: In (d), averaging the two means (501 + 500.2)/2 ignores the different group sizes and scores no marks.

Where marks are usually lost

• Writing “outlier” without the numerical boundaries Q₁ − 1.5 × IQR and Q₃ + 1.5 × IQR.
• Drawing whiskers to outliers, or not marking outliers with a cross.
• Estimating a grouped mean with class boundaries instead of mid-interval values.
• Forgetting to subtract from the total for “more than” on a cumulative frequency graph.
• Changing the standard deviation when a constant is added, or multiplying the variance by k instead of k².
• Comparing distributions without context, or without both centre and spread.
• Averaging two group means without weighting by group size.

Next steps

• Revision notes for this unit
• Study guide for this unit
• IB DP Mathematics: Applications and Interpretation course hub
• Printable syllabus checklist
• All free 10-minute diagnostics
• Book a free trial class

Official syllabus

International Baccalaureate Organization, Diploma Programme, Mathematics: applications and interpretation guide, first assessment 2021 – syllabus sections 4.1, 4.2 and 4.3.