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Mathematics questions, chapter 7
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Measures of Dispersion for Grouped Data

42 marks. Use Practice to check each answer as you go, or Exam to answer everything first. Show your working for Section B. A scientific calculator is allowed.

Section A: objective questions

13 marks
Choose the correct answer. Each question carries 1 mark.
1

Question 1

A frequency table has the classes 20 – 24 and 25 – 29. What is the lower boundary of the class 25 – 29?
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B
Lower boundary = (24 + 25) ÷ 2 = 24.5. It is halfway between the upper limit of the class before and the lower limit of this class.
For the teacher: Remembering, low difficulty, standard 7.1.1
Textbook p. 199
2

Question 2

The classes of a frequency table for heights in metres are 1.0 – 1.9, 2.0 – 2.9, 3.0 – 3.9, … What is the size of each class interval?
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C
Size = difference between two consecutive lower limits = 2.0 − 1.0 = 1.0. Using 1.9 − 1.0 = 0.9 (limits of one class) is the common mistake.
For the teacher: Understanding, low difficulty, standard 7.1.1
Textbook p. 200
3

Question 3

The frequencies of four consecutive classes are 4, 9, 13 and 8. What is the cumulative frequency of the third class?
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B
4 + 9 + 13 = 26. 34 is the cumulative frequency of the fourth class; 22 leaves out the first class.
For the teacher: Applying, low difficulty, standard 7.1.1
Textbook p. 200
4

Question 4

In an ogive, the cumulative frequency is plotted against the ______ of each class.
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D
An ogive uses the upper boundary because the cumulative frequency counts all the values up to that boundary.
For the teacher: Remembering, low difficulty, standard 7.1.3
Textbook p. 207
5

Question 5

When drawing a frequency polygon, a class with zero frequency is added before the first class and after the last class. Why?
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A
The two extra midpoints with frequency 0 let the polygon close at the axis. They do not change the total frequency or the data.
For the teacher: Understanding, medium difficulty, standard 7.1.1
Textbook p. 202
6

Question 6

Diagram 1 shows the time taken by pupils to travel to school. What is the shape of the distribution?
Histogram with six bars of heights 12, 18, 10, 6, 3 and 1: it peaks at the second bar and has a long tail on the right
Diagram 1
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C
Most pupils have short travel times and the long tail is on the right (the larger values), so it is skewed to the right. Skewness is named after the tail, not the peak.
For the teacher: Understanding, low difficulty, standard 7.1.2
Textbook p. 204
7

Question 7

An ogive is drawn for 120 data. At which cumulative frequency is the third quartile Q3 read?
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C
Position of Q3 = ¾ × 120 = 90. 30 is for Q1 and 60 is for the median.
For the teacher: Applying, medium difficulty, standard 7.1.3
Textbook p. 208
8

Question 8

An ogive is drawn for 200 data. At which cumulative frequency is the 35th percentile, P35, read?
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B
35% of 200 = 0.35 × 200 = 70. 35 is the percentile itself, not the position.
For the teacher: Applying, medium difficulty, standard 7.1.3
Textbook p. 209
9

Question 9

A frequency table has the classes 10 – 19, 20 – 29, 30 – 39, 40 – 49 and 50 – 59. What is the range of the data?
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A
Range = midpoint of the highest class − midpoint of the lowest class = 54.5 − 14.5 = 40. 49 comes from 59 − 10, which uses the limits.
For the teacher: Applying, medium difficulty, standard 7.2.1
Textbook p. 211
10

Question 10

For 50 data, Σfx = 2 000 and Σfx² = 82 000. What is the standard deviation?
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B
Mean = 2 000 ÷ 50 = 40. Variance = 82 000 ÷ 50 − 40² = 1 640 − 1 600 = 40. Standard deviation = √40 = 6.32. 40 is the variance, not the standard deviation.
For the teacher: Applying, medium difficulty, standard 7.2.1
Textbook p. 212
11

Question 11

Diagram 2 shows a box plot of the marks of a group of pupils. Which statement describes the distribution?
Box plot: minimum 10, Q1 40, median 52, Q3 58, maximum 66. The left part of the box and the left whisker are longer than the right ones
Diagram 2
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D
The left whisker (10 to 40) and the left part of the box (40 to 52) are longer than the right ones (58 to 66 and 52 to 58), so it is skewed to the left.
For the teacher: Analysing, medium difficulty, standard 7.2.2
Textbook p. 213
12

Question 12

Set A has mean 68 and standard deviation 4.2. Set B has mean 68 and standard deviation 7.5. Which conclusion is correct?
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A
With equal means, the set with the smaller standard deviation has its values closer to the mean, so it is more consistent.
For the teacher: Evaluating, high difficulty, standard 7.2.3
Textbook p. 215
13

Question 13

The classes 0 – 9, 10 – 19 and 20 – 29 have frequencies 4, 10 and 6. What is the mean?
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D
Midpoints 4.5, 14.5, 24.5. Σfx = 18 + 145 + 147 = 310. Mean = 310 ÷ 20 = 15.5.
For the teacher: Applying, high difficulty, standard 7.2.1
Textbook p. 212

Section B: structured questions

29 marks
Answer all questions. Show your working.
14

Question 14

Diagram 3 is an ogive of the time spent on homework by 80 pupils in one night.
Ogive of 80 pupils with points (9.5, 0), (19.5, 4), (29.5, 20), (39.5, 40), (49.5, 60), (59.5, 74) and (69.5, 80)
Diagram 3
[3 marks] (a)Find the first quartile, the median and the third quartile. (Accept values within 1 minute.)
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Tick each point you got.
[2 marks] (b)Hence find the interquartile range and say what it means.
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Tick each point you got.
[2 marks] (c)Find the 90th percentile, P90. (Accept 57 to 59.)
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Tick each point you got.
[1 mark] (d)What percentage of the pupils spent 44.5 minutes or less?
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Tick each point you got.
For the teacher: Applying, medium difficulty, standard 7.1.3
Textbook p. 208
15

Question 15

The table shows the time, in seconds, taken by 40 swimmers to complete a lap.
Time (s)20 – 2930 – 3940 – 4950 – 5960 – 69
Number of swimmers591484
[2 marks] (a)Calculate the mean time.
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Tick each point you got.
[2 marks] (b)Calculate the variance, correct to 2 decimal places.
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Tick each point you got.
[1 mark] (c)Hence find the standard deviation.
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Tick each point you got.
[1 mark] (d)Find the range of the data.
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Tick each point you got.
For the teacher: Applying, medium difficulty, standard 7.2.1
Textbook p. 212
16

Question 16

Two classes, A and B, took the same Mathematics test (30 pupils each).
Marks11 – 2021 – 3031 – 4041 – 5051 – 60
Class A261273
Class B148107
[2 marks] (a)Find the mean mark of each class.
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Tick each point you got.
[2 marks] (b)Find the standard deviation of each class, correct to 2 decimal places.
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Tick each point you got.
[2 marks] (c)Which class did better and which class was more consistent? Justify your answer.
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Tick each point you got.
For the teacher: Evaluating, high difficulty, standard 7.2.3
Textbook p. 215
17

Question 17

Diagram 4 is a box plot of the homework times of the same 80 pupils as in Diagram 3.
Box plot with minimum 9.5, Q1 29.5, median 39.5, Q3 49.5 and maximum 69.5 minutes
Diagram 4
[2 marks] (a)State the five values shown on the box plot.
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Tick each point you got.
[1 mark] (b)State the interquartile range.
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Tick each point you got.
[2 marks] (c)Describe the shape of the distribution and give a reason.
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Tick each point you got.
For the teacher: Analysing, medium difficulty, standard 7.2.2
Textbook p. 214
18

Question 18

Your class will investigate the daily time spent on screens by Form 5 pupils in your school, and report with measures of central tendency and dispersion.
[2 marks] (a)Describe how you would choose 40 respondents and state one question you would ask.
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Tick each point you got.
[2 marks] (b)Name two graphs and two measures you would use, and say why.
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Tick each point you got.
For the teacher: Creating, high difficulty, standard 7.2.5
Textbook p. 219