Statistics – Averages and Analysis — Free IB MYP3 Mathematics Questions

1QuestionCalculating Midpoints for IntervalsConcept Practice

3 marks~5 minCriterion B

A survey records the ages of visitors to a museum. The grouped data are shown below.

Interval	0–10	10–20	20–30	30–40	40–50
Midpoint	5	15	25	35	45

a

Describe the pattern in the midpoints shown in the table. [1]

b

Calculate the midpoint of the interval 50–60. Show your method. [1]

c

A new interval has a midpoint of 75. Explain what the interval must be, and justify your answer using the midpoint formula. [1]

Solutions

2QuestionCalculating Midpoints for IntervalsConcept Practice

2 marks~3 minCriterion A

A teacher records student test scores in equal intervals: 0–10, 10–20, 20–30, 30–40, and 40–50.

a

Calculate the midpoint of the interval 20–30. [1]

b

Explain why statisticians use midpoints to represent all scores within an interval when working with grouped data. [1]

Solutions

3QuestionUsing the Mean to Find a Missing ValueConcept Practice

2 marks~3 minCriterion B

Each set below contains five numbers with a mean of 10. One number in each set is missing.

Set 1:

8,\ 12,\ 7,\ 13,\ ?

Set 2:

9,\ 11,\ 10,\ 8,\ ?

Set 3:

15,\ 5,\ 12,\ 6,\ ?

Set 4:

4,\ 14,\ 9,\ 13,\ ?

a

Calculate the missing number in each set. [1]

b

Explain a rule that can always be used to find a missing value when the mean and the number of values are known. [1]

Solutions

4QuestionCreating Balanced Data SetsConcept Practice

2 marks~3 minCriterion A

Five bags of marbles contain the following numbers of marbles:

Bag 1: 3 — Bag 2: 5 — Bag 3: 7 — Bag 4: 9 — Bag 5:

x

The mean number of marbles per bag is 6.

Calculate the value of

x

. [2]

Solutions

5QuestionCalculating and Interpreting the Interquartile RangeConcept Practice

2 marks~3 minCriterion A

The box plot below shows the distribution of test scores for a class.

Minimum: 40 —

Q_1

: 55 — Median: 70 —

Q_3

: 85 — Maximum: 100

Calculate the interquartile range (IQR) of the test scores. [2]

IQR = Q_3 - Q_1

Solutions

6QuestionCommon Errors in Average InterpretationConcept Practice

2 marks~3 minCriterion A

The bar chart below shows the favourite sports of students in a class.

Football: 8 students
Basketball: 5 students
Swimming: 3 students
Tennis: 5 students

a

State the mode of the favourite sports data. [1]

b

Explain why Football is the mode and not Basketball or Tennis, even though both have the same frequency as each other. [1]

Solutions

7QuestionCalculating the Range and Interpreting ItConcept Practice

2 marks~3 minCriterion A

Five line segments have the following lengths (in cm):

3, 7, 12, 4, 9

a

Calculate the range of these lengths. [1]

b

Explain what the range indicates about the spread of the five measurements. [1]

Solutions

8QuestionAnalyzing Real-Life Data with AveragesConcept Practice

2 marks~3 minCriterion A

The frequency table below shows the number of books read by students in a class last month.

Number of books	0	1	2	3
Number of students	3	5	4	2

Calculate the mean number of books read per student. Show your working clearly. [2]

Solutions

9QuestionAnalyzing Real-Life Data with AveragesConcept Practice

2 marks~3 minCriterion B

A student records their study time and test scores.

Hours studied ( $h$ )	2	4	6	8
Test score ( $s$ )	55	65	75	85

a

Describe the pattern relating hours studied to test score. [1]

b

Calculate the predicted test score if the student studies for 10 hours. [1]

Solutions

10QuestionGrouping Data into Class IntervalsAssessment Practice

6 marks~9 minCriterion C

A school records the ages of 200 students in a sports club. Ages range from 6 to 18 years. The school groups all the data into one class interval:

6–18

years.

a

Describe what information is hidden when all 200 students are placed into a single class interval. [2]

b

Propose a set of class intervals for this data and explain how your intervals improve understanding of the age distribution. [2]

c

Analyse one limitation of using class intervals to represent this data, even when the intervals are well chosen. [2]

Solutions

11QuestionEstimating Mean from Grouped DataAssessment Practice

2 marks~3 minCriterion D

A school records the heights of 100 students in the following grouped frequency table.

Height (cm)	140–150	150–160	160–170	170–180	180–190
Frequency	10	30	35	20	5

a

Identify a specific real-world scenario in which estimating a mean from grouped height data would be useful. [1]

b

Explain one limitation of using an estimated mean in the scenario you identified. [1]

Solutions

12QuestionCreating Balanced Data SetsAssessment Practice

6 marks~9 minCriterion D

A basketball coach records the points scored by five players in one game:

Player A: 14 — Player B: 10 — Player C: 16 — Player D: 8 — Player E: unknown

To qualify for a tournament, the team's mean score must be exactly 13 points.

a

Calculate the score Player E must achieve to reach this mean. [2]

b

Explain why using the mean is a useful way to set a team performance target. [2]

c

Analyse whether it is realistic to rely on one player's score to guarantee the team's mean. In your answer, refer to at least two limitations of the mean in this context. [2]

Solutions

13QuestionCreating Balanced Data SetsAssessment Practice

5 marks~8 minCriterion C

A school basketball team played 5 games. Their points scored in each game were:

8, \ 10, \ 15, \ x, \ y

. The sports coordinator claims the mean points per game is exactly

12

.

a

Calculate two possible values for

x

and

y

. Show your working. [2]

b

Explain one limitation of using only the mean to assess the team's performance. [1]

c

The coordinator compares two possible score sets:

Set A:

8, \ 10, \ 15, \ 13, \ 14

Set B:

8, \ 10, \ 15, \ 1, \ 26

Both sets have a mean of

12

. Analyse which set better represents a consistent team performance, and justify your answer using a relevant statistical measure. [2]

---

Solutions

14QuestionDrawing Box Plots from Data SetsAssessment Practice

6 marks~9 minCriterion B

Five data sets, each with a median of 10, are shown below.

Data Set A:

2, 6, 10, 14, 18

(range

= 16

)
Data Set B:

4, 7, 10, 13, 16

(range

= 12

)
Data Set C:

6, 8, 10, 12, 14

(range

= 8

)
Data Set D:

8, 9, 10, 11, 12

(range

= 4

)
Data Set E:

0, 5, 10, 15, 20

(range

= 20

)

a

Calculate the IQR for each data set. Show your working. [2]

b

Describe the relationship between the range and the IQR for these data sets, and state a general rule linking IQR to range for symmetric, equally spaced sets of five values. [2]

c

A new data set has the same structure, a median of 10, and a range of 24. Calculate the IQR you would expect, then verify your answer by finding

Q_1

and

Q_3

for the data set

-2, 4, 10, 16, 22

. [2]

Solutions

15QuestionComparing Data Sets Using Box PlotsAssessment Practice

6 marks~9 minCriterion D

Two mathematics classes took the same test. Their results are shown in the box plots.

Class A	median $= 72$	$\text{IQR} = 18$	range $= 45$
Class B	median $= 68$	$\text{IQR} = 10$	range $= 25$

a

State the median score for each class. [1]

b

Compare the IQRs of the two classes. Explain what the difference in IQRs indicates about the consistency of scores in each class. [2]

c

The principal wants to award a prize to the class that performed better overall. Analyse whether the median alone is sufficient to make this decision, using the IQR and range to support your answer. [3]

Solutions

16QuestionComparing Data Sets Using Box PlotsAssessment Practice

5 marks~8 minCriterion C

The box plots below show the mathematics exam scores for Class A and Class B.

Class A	Minimum 45	$Q_1 = 55$	Median $= 65$	$Q_3 = 75$	Maximum 90
Class B	Minimum 50	$Q_1 = 60$	Median $= 70$	$Q_3 = 80$	Maximum 95

a

Calculate the interquartile range (IQR) for each class. [2]

b

Compare the medians of the two classes. Explain what this tells you about overall performance. [1]

c

A student claims: "Class B performed better overall and had more consistent scores." Using your results from (a) and (b), analyse whether this claim is fully correct, partially correct, or incorrect. Justify your answer. [2]

Solutions

17QuestionRecognizing Misleading GraphsAssessment Practice

8 marks~12 minCriterion B

A small business records monthly sales (in thousands of dollars) for January to April: 50, 55, 60, 65.

Three bar charts display these data using different y-axis starting values:
Chart A starts at 0, Chart B starts at 40, Chart C starts at 48.

The bar height for any month

=

actual value

-

y-axis start.

a

Calculate the perceived percentage increase from January to April for Charts A and B, using:

\text{Perceived \% increase} = \frac{\text{April bar height} - \text{January bar height}}{\text{January bar height}} \times 100\%

[2]

b

Explain how raising the y-axis start value affects the visual impression of the increase shown in the charts. [2]

c

Analyse Chart C: using the formula from part (a), calculate the perceived percentage increase, then compare it with the actual percentage increase and explain what this reveals about the effect of a truncated y-axis. [4]

Solutions

18QuestionManipulating Data for PersuasionAssessment Practice

6 marks~9 minCriterion C

A toothpaste advertisement claims: "Preferred by 4 out of 5 dentists!" The company surveyed only 10 dentists, all of whom were paid consultants for the company.

a

Describe what is meant by a representative sample and explain why this survey may not produce one. [2]

b

Explain how reporting

\dfrac{4}{5}

(80%) rather than "8 out of 10 dentists" could mislead consumers. [2]

c

A consumer sees the advertisement and says: "80% of dentists prefer it — that's strong evidence!" Analyse this statement using the survey's sample size and data collection method. [2]

Solutions

19QuestionCommon Errors in Average InterpretationAssessment Practice

6 marks~9 minCriterion D

A company claims that the average salary at their firm is

75\,000

dollars per year. Further investigation reveals that the median salary is

52\,000

dollars and the mode is

45\,000

dollars.

a

Identify which measure of central tendency the company is most likely using to support their claim. [1]

b

Explain why the company might prefer this measure over the median or the mode. [2]

c

Analyse the limitations of using a single average to represent the salary data, and suggest two pieces of additional information that would give a more complete picture of the distribution. [3]

Solutions

20QuestionCalculating the Range and Interpreting ItAssessment Practice

6 marks~9 minCriterion B

Consider the data set:

2, 5, 8, 11, 14

.

a

Calculate the range of the data set. Then add one new number so that the range increases by exactly 6. Explain your reasoning. [2]

b

Using your new set from part (a), explain which type of number you would add to make the range as large as possible, and why. Add a number of your choice and calculate the new range. [2]

c

Analyse how adding a value outside the current range affects the range of a data set. Choose a different starting set, apply your rule, and show that it holds. [2]

Solutions

21QuestionDetermining the Mode in a Data SetAssessment Practice

4 marks~6 minCriterion D

A small business owner records the number of customers per day over two weeks:

3, \ 5, \ 5, \ 5, \ 5, \ 5, \ 5, \ 5, \ 5, \ 5, \ 5, \ 5, \ 5, \ 50

The owner claims that the "typical" number of daily customers is

5

because it is the mode.

a

State the mode of the data set. [1]

b

Explain what the mode indicates about the daily customer count in this context. [1]

c

Analyse whether the mode is a reliable measure of the typical number of daily customers, referring to the outlier in the data set. [2]

Solutions

22QuestionCalculating the Range and Interpreting ItAssessment Practice

4 marks~6 minCriterion C

A basketball coach records the points scored by two players over five games.

Player A	12	15	14	13	16
Player B	8	22	10	9	21

a

Calculate the range of points scored by each player. [2]

b

Explain one limitation of using the range to compare the consistency of the two players, referring to Player B's data in your answer. [2]

Solutions

23QuestionChoosing the Most Appropriate AverageAssessment Practice

2 marks~3 minCriterion D

A small business owner records daily sales (in dollars):

110, 120, 130, 135, 500

The owner must report a figure for typical daily sales to a bank for a loan application.

a

Calculate the mean and median of the sales data. [1]

b

Explain which average is more appropriate to report to the bank, and how using the other average could mislead the bank. [1]

Solutions

24QuestionChoosing the Most Appropriate AverageAssessment Practice

6 marks~9 minCriterion C

A small company has seven employees with the following annual salaries (thousands of dollars):

Employee	A	B	C	D	E	F	G
Salary	30	32	35	38	40	42	120

The owner quotes the mean salary in job advertisements. Employees argue the median is fairer.

a

Calculate the mean and median salary. [2]

b

Explain why the median is more appropriate than the mean for representing a typical employee's salary. [2]

c

Analyse one limitation of using a single average to describe the salary distribution in this company. [2]

Solutions

Statistics – Averages and Analysis — Free MYP3 Mathematics Practice Questions