4.3 Measures of central tendency and dispersion questions for IB Maths AI

56 questions on Measures of central tendency and dispersion, written by Augment in the style of IB exam papers. Six of them are below in full, to answer on paper.

IB Mathematics: Applications and Interpretation
Part of 4 Statistics and probability
SL and HL

Questions by paper and level

SL and HL questions are for students at either level. HL only questions cover what only HL students study.

Paper	SL and HL	HL only	All
Paper 1	22	1	23
Paper 2	31	1	32
Paper 3HL only	0	1	1
Every paper	53	3	56

Counted on 10 November 2026

Questions to try

Written by Augment in the style of IB exams, not past papers. Answer them on paper, then check each one against its markscheme with a free account.

Question 1 · AI-00193Paper 1 · 6 marks

A sports scientist measures the resting heart rate, in beats per minute (bpm), of 11 rowers in a university squad. The results are:

48, 52, 55, 56, 57, 58, 60, 61, 63, 64, 82

(a)
Write down the value of
(i) the lower quartile;
(ii) the upper quartile.
2 marks
(b)
Show that the value 82 bpm is an outlier.
3 marks
(c)
The scientist later finds that the reading of 82 bpm was taken from a rower who had just finished a warm-up. State, with a reason, whether this value should be removed from the data.
1 mark

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Question 2 · AI-00198Paper 2 · 18 marks

Lindiwe's farm has three apple orchards. The numbers of apple trees in each orchard are shown in the following table.

Orchard	Hillside	Riverbank	Valley
Number of trees	1840	1150	1010

Lindiwe wants to check 80 trees for disease.

(a)
She decides to use a stratified sample. Find the number of trees she should check in each orchard.
3 marks
(b)
Explain why a stratified sample may be more suitable than a simple random sample of 80 trees.
1 mark
(c)
Lindiwe also weighs apples. State whether the mass of an apple is discrete or continuous data.
1 mark
(d)
The masses, in grams, of 12 apples picked from Riverbank are:
$142, 156, 161, 148, 170, 152, 165, 139, 158, 147, 163, 236$
Find
(i) the mean mass;
(ii) the standard deviation of the masses.
3 marks
(e)
(i) Find the lower quartile, the median and the upper quartile of the masses.
(ii) Show that the mass of 236 g is an outlier.
6 marks
(f)
Draw a box-and-whisker diagram to represent the masses of the 12 apples, using a horizontal scale from 120 g to 240 g.
3 marks
(g)
Lindiwe checks her notebook and confirms that the 236 g apple was weighed correctly. State, with a reason, whether she should remove it from the data.
1 mark

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Question 3 · AI-01642Paper 3 · HL only · 27 marks

This question investigates whether the opening of a new marina has increased the number of boats arriving at a canal lock.

Boats on the Ellesmere Canal must pass through Hartwell Lock. The lock keeper, Ingrid, records the number of boats arriving at the lock in 40 randomly chosen one-hour periods during the summer.

Number of boats in one hour	0	1	2	3	4	5	6	7
Frequency	2	4	9	9	7	5	3	1

(a)
Ingrid wants to model the number of boats arriving in one hour by a Poisson distribution.
State two conditions that must hold for this model to be appropriate.
2 marks
(b)(i)
Calculate the mean and the variance of the number of boats arriving in one hour, according to Ingrid's data.
2 marks
(b)(ii)
Explain how your answers to part (b)(i) support the use of a Poisson model.
1 mark
(c)(i)
Ingrid models the number of boats arriving in one hour by a Poisson distribution with mean 3.2.
Find the probability that exactly 5 boats arrive in a one-hour period.
2 marks
(c)(ii)
Given that at least 3 boats arrive in a two-hour period, find the probability that fewer than 6 boats arrive in that period.
3 marks
(d)(i)
The lock operates in 30-minute cycles and can take at most 4 boats in each cycle. Any extra boats must wait for the next cycle.
Find the probability that more than 4 boats arrive during one 30-minute cycle.
2 marks
(d)(ii)
On one day the lock operates 20 cycles. The numbers of boats arriving in different cycles are independent.
Find the probability that more than 4 boats arrive in at least two of the cycles on that day.
3 marks
(e)(i)
The following summer a new marina opens upstream. Ingrid believes that the mean number of boats arriving per hour has increased. She will count the number of boats, $T$ , arriving during one randomly chosen five-hour period and test her belief at the 5 % significance level.
State the null and alternative hypotheses for this test.
1 mark
(e)(ii)
Find the critical region for this test.
3 marks
(e)(iii)
During the five-hour period, 25 boats arrive at the lock.
State the conclusion of the test, in context. Give a reason for your answer.
2 marks
(f)
Find the probability of a Type I error for this test.
2 marks
(g)
In fact, after the marina opens, the mean number of boats arriving is 4.4 per hour.
Find the probability of a Type II error for this test.
3 marks
(h)
Suggest how Ingrid could reduce the probability of a Type II error without increasing the significance level of the test.
1 mark

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Question 4 · AI-00194Paper 1 · 7 marks

In a biology class, ten students each measure the height of one bean seedling, in cm, after three weeks of growth. Their results are recorded in a shared table.

Student	1	2	3	4	5	6	7	8	9	10
Height (cm)	12.4	11.8	13.1		12.9	118	12.2	13.5	11.9	12.6

Student 4 was absent and did not record a height.

(a)
Identify the value that has been recorded in error. Suggest what is likely to have happened.
2 marks
(b)
The teacher removes the value in error. Find the mean height of the remaining eight seedlings.
2 marks
(c)
Another student suggests entering 0 for student 4. Explain why this is not appropriate.
1 mark
(d)
The teacher checks with student 6, who confirms that the height was 11.8 cm. Find the median height of the nine recorded seedlings.
2 marks

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Question 5 · AI-00199Paper 2 · 14 marks

The council of the town of Dunmore wants to find out how long residents take to travel to work. A council officer stands at the central bus station one morning and asks 50 people waiting there how long their journey to work takes.

(a)
Name the sampling technique used by the officer.
1 mark
(b)
Explain why this sample is likely to be biased.
1 mark
(c)
The council has a list of all 18 400 adult residents. Describe how a simple random sample of 120 residents could be chosen from this list.
2 marks
(d)
The council takes a simple random sample of 120 residents. The time, $t$ minutes, that each takes to travel to work is shown in the following table.
Time ( $t$ minutes)
$0 < t \leq 10$
$10 < t \leq 20$
$20 < t \leq 30$
$30 < t \leq 40$
$40 < t \leq 50$
$50 < t \leq 60$
Number of residents
14
31
38
22
11
4
Write down the modal class.
1 mark
(e)
Use the mid-interval values to find an estimate of
(i) the mean time;
(ii) the standard deviation of the times.
4 marks
(f)
Explain why your answer to part (e)(i) is only an estimate.
1 mark
(g)
Find the class that contains the median time.
2 marks
(h)
The council's target is that no more than 15% of residents travel for more than 40 minutes. Determine whether the sample suggests that the target is being met.
2 marks

Time ( $t$ minutes)	$0 < t \leq 10$	$10 < t \leq 20$	$20 < t \leq 30$	$30 < t \leq 40$	$40 < t \leq 50$	$50 < t \leq 60$
Number of residents	14	31	38	22	11	4

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Question 6 · AI-00376Paper 1 · 7 marks

The times taken by 160 runners to complete a 5 km park run are shown on the following cumulative frequency curve.

Cumulative frequency curve of the times of 160 runners. Horizontal axis: time in minutes from 16 to 40. Vertical axis: cumulative frequency from 0 to 160. The curve passes through (16, 0), (20, 12), (24, 46), (28, 98), (32, 134), (36, 152) and (40, 160).

(a)
Use the cumulative frequency curve to estimate
(i) the median time;
(ii) the lower quartile;
(iii) the upper quartile;
(iv) the interquartile range.
4 marks
(b)
Estimate the number of runners who took more than 30 minutes.
2 marks
(c)
Write down the number of runners who took between 24 and 28 minutes.
1 mark

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