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10.06 Cumulative frequency tables and graphs

Worksheet
Cumulative frequency tables
1

Consider the following table:

a

Find the total number of scores recorded.

b

Find the number of times a score of 14 occurred.

c

Find the number of times a score less than 13 occurred.

\text{Score } (x)\text{Cumulative} \\ \text{frequency } (cf)
107
1115
1218
1320
1426
2

Complete the given frequency table:

\text{Score } \left(x\right)\text{Frequency} \\ \left(f\right)\text{Cumulative} \\ \text{frequency} \\ \left(cf\right)
1288
135
14821
1526
3

For the given frequency table:

a

Complete the cumulative frequency column.

b

Calculate the total frequency.

c

State the class size.

\text{Score } \left(x\right)\text{Frequency } \\ \left(f\right)\text{Cumulative} \\ \text{frequency} \\ \left(cf\right)
1 \to 44
5 \to 85
9 \to 129
13 \to 165
17 \to 204
4

For the given frequency table:

a

Complete the cumulative frequency column.

b

Calculate the total frequency.

c

State the class size.

d

Approximately half of the scores recorded are greater than what score?

\text{Score } \left(x\right)\text{Frequency } \\ \left(f\right)\text{Cumulative} \\ \text{frequency} \\ \left(cf\right)
1 \to 515
6 \to 1026
11 \to 1518
16 \to 2014
21 \to 257
26 \to 302
5

For the given frequency table:

a

Complete the cumulative frequency column.

b

Calculate the total frequency.

c

State the class size.

d

Approximately one third of the scores recorded are greater than what score?

\text{Score } \left(x\right)\text{Frequency } \\ \left(f\right)\text{Cumulative} \\ \text{frequency} \\ \left(cf\right)
20 \to 247
25 \to 2918
30 \to 3425
35 \to 3912
40 \to 448
45 \to 494
50 \to 541
Cumulative frequency graphs
6

Construct a cumulative frequency table for the data represented in the given histogram:

7

Construct a frequency table for the data represented in the given cumulative frequency histogram:

8

Consider the following cumulative frequency histogram:

a

Find the total number of scores recorded.

b

Find the number of times a score of 46 occurred.

c

Find the number of times a score of 45 occurred.

d

Find the percentage of scores that were 43 or less. Round your answer to one decimal place.

9

Consider the frequency table showing the number of 'holes in one' across golf tournaments:

a

Construct a cumulative frequency histogram for this data.

b

Find the total number of 'holes in one' across all the tournaments.

c

In how many tournaments were at most 3 'holes in one' scored?

\text{Number of } \\ \text{holes in one}\text{Tournaments}
25
31
43
54
60
10

The following graph shows the cumulative frequency ogive of the masses of fish caught in a fishing competition:

Construct a frequency histogram to depict the distribution of the masses of the fish.

50
100
150
200
250
300
350
\text{Mass (g)}
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
\text{Cumulative frequency}
11

The marks received by a group of 20 students are given below:

55,\, 90,\, 90,\, 99,\, 78,\, 78,\, 78,\, 99,\, 78,\, 99,\, 55,\, 99,\, 78,\, 64,\, 64,\, 78,\, 78,\, 99,\, 64,\, 78

a

Construct a cumulative frequency histogram to represent the data.

b

How many students scored above 70?

c

How many students scored less than or equal to 70?

d

How many students scored above 80 but less than or equal to 90?

Applications
12

A principal wants to investigate the performance of students at his school in Performing Arts. To do this, he has the marks of each student studying Performing Arts collected into groups and put into a frequency table. Each group of marks is assigned a grade as shown in the following frequency table:

\text{Grade}\quad \text{Score } \left(x\right) \quad\text{Frequency } \left(f\right)\text{Cumulative frequency } \left(cf\right)
\text{E}0 \leq x < 207
\text{D}20 \leq x < 4014
\text{C}40 \leq x < 6032
\text{B}60 \leq x < 8097
\text{A}80 \leq x < 10062
a

Complete the table by finding the cumulative frequency values.

b

Calculate the total frequency.

c

State the class size.

d

Approximately three quarters of the scores recorded are greater than what score?

13

A pair of dice are rolled 50 times and the numbers appearing on the uppermost face are added to give a score. The results are recorded in the given table:

a

State the lowest possible score when a single pair of dice are rolled.

b

State the highest possible score when a single pair of dice are rolled.

c

Complete the table by finding the cumulative frequency values.

d

Find the number of times a score of 8 occurred.

e

Find the number of times a score more than 9 occurred.

f

Find the number of times a score of at most 6 occurred.

\text{Score} \\ (x)\text{Frequency} \\ (f)\text{Cumulative} \\ \text{frequency } (cf)
21
32
45
55
65
79
87
95
108
111
122
14

The number of sightings of the Northern Lights were recorded across various Canadian locations over a period of 1 month. The list below represents the number of sightings at each location:

11,\, 10,\, 10,\, 9,\, 7,\, 8,\, 8,\, 12,\, 12,\, 12,\, 12,\, 12,\, 12,\, 9,\, 9,\, 12,\, 9,\, 9,\, 8,\, 8

a

Complete the table.

b

In how many locations were there at least 8 sightings?

c

In how many locations were there less than 11 sightings?

\text{Number of} \\ \text{sightings}\text{Number of} \\ \text{locations } \left(f\right)\text{Cumulative} \\ \text{frequency } \left(cf\right)
7
8
9
10
11
12
15

A 1500 \text { m} swimmer records her time over several training sessions. Her times are recorded in the following histogram:

a

Construct a cumulative frequency table for the data using the given intervals.

b

Find the total number of training sessions she completed.

c

Find the number of times she recorded a swim time faster than 16:40.

d

Find the percentage of swims that were less than 16:30.

16

The heights of 22 boys in a class are listed:

164, \, 167, \, 158, \, 159, \, 150, \, 166, \, 150, \, 146, \, 149, \, 161, \, 164, \\ 163, \, 152, \, 161, \, 157, \, 157, \, 153, \, 157, \, 156, \, 165, \, 162, \, 161

a

Construct a cumulative frequency histogram for the data. Use the discrete intervals of 146-150,\, 151-155, etc.

b

How many students are taller than 155 \text{ cm}?

c

How many students are at most 155 \text{ cm} tall?

d

How many students are taller than 150 \text{ cm} but shorter than 156 \text{ cm}?

e

Calculate the centre of the class 151 - 155.

17

A pair of dice are rolled 50 times and the numbers appearing on the uppermost face are added to give a score. The results are presented in the following histogram:

a

Explain how the frequency of a certain score can be determined from a cumulative frequency histogram.

b

State the most frequent score.

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Outcomes

MS11-2

represents information in symbolic, graphical and tabular form

MS11-7

develops and carries out simple statistical processes to answer questions posed

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