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Name: ___________________
(Sec 4) Statistical Data Analysis - Cumulative Frequency Tables and
Curves
1. A survey was conducted and participants were asked about the number of hours of
sleep on average (t) they get every night. Each participant is allowed to respond to
survey only once. The data is collected and shown in the cumulative frequency table
below.
(i) How many people participated in the survey?
(ii) Using a scale of 1 cm to represent 1 hour on the horizonal axis and 1 cm to
represent 10 students on the vertical axis, draw the cumulative frequency curve
for the given data.
(iii) Use your curve in (ii) to estimate
(a) the number of adults who get 6.5 hours or less of sleep on average,
(b) the number of adults who sleep for more than 8.5 hours,
(c) the value of t, such that 75% of the adults surveyed sleep for at least t
hours or less.
No. of Hours, t Cumulative Frequency
3t 0
4t 1
5t 7
6t 22
7t 38
8t 62
9t 75
10t 80
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2. The lengths of 100 mudskippers were measured and recorded in the frequency table
below.
Length (cm), x Frequency
12x 0
12 14x 10
14 16x 18
16 18x 30
18 20x 30
20 22x 10
22 24x 2
(i) Construct a table of cumulative frequencies for the given data.
(ii) Using a scale of 1 cm to represent 2 cm on the horizonal axis and 1 cm to
represent 10 mudskippers on the vertical axis, draw the cumulative frequency
curve for the given data.
(iii) The average length of a mudskipper is normally 17 cm. Use your curve in (ii)
to calculate the percentage of this group that is of average size or below.
(iv) Calculate an estimate for the mean length of this particular group of
mudskippers.
(v) Based on your estimate, what can you say about the lengths of this group of 100
mudskippers compared to the normal average of 17 cm?
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3. The speeds of 70 Personal Mobility Devices (PMDs) on a footpath were recorded.
Their speeds were distributed as shown in the cumulative frequency curve below.
(i) From the graph, estimate
(a) the number of PMDs with speeds less than or equal to 10 km/h,
(b) the number of PMDs with speeds less than or equal to 15 km/h,
(c) the percentage of PMDs with speeds greater than 25 km/h.
(ii) A PMD is selected at random from the group. Find the probability that the speed
of the randomly selected PMD is
(a) greater than 25 km/h,
(b) either not greater than 10 km/h or greater than 25 km/h.
(iii) If two PMDs are chosen at random from the group, find the probability that the
speeds of both PMDs are greater than 15 km/h.
Speed (km/h)
15
50
40
30
20
10
0 5 10
60
20 25
70 C
um
ula
tive
Fre
qu
ency
30
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4. The following cumulative frequency table shows the average time (t) spent weekly on
the social media by a group of 100 students surveyed.
No. of Hours, t Cumulative Frequency
2t 8
4t 18
6t 34
8t 55
10t 74
12t 89
14t 100
(i) Construct a frequency table for the given data.
(ii) Using yours answer in (i), draw a histogram to represent the frequency
distribution in the grid below.
(iii) Determine
(a) the modal class,
(b) the percentage of students who spend more than 6 hours a week on social
media,
(c) the percentage of students who spend more than 4 but less than or equal
to 10 hours a week on social media,
(d) the mean amount of time spent on social media.
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5. The number of hours spent watching Netflix by a group of families in a week is shown
in the cumulative frequency curve below.
(i) Assuming each family was only surveyed once, how many families were
involved?
(ii) Use the curve to estimate
(a) the number of families who watch 15 hours or less a week,
(b) the percentage of families who watch more than 35 hours a week,
(c) the value of x, if 30% of families watch x hours or less a week.
(iii) A family is randomly selected from the group. Find the probability that they
watch
(a) equal to or less than x hours a week,
(b) more than 25 hours a week,
(c) more than 20 hours but less than or equal to 30 hours a week.
Cu
mu
lati
ve
Fre
qu
ency
Time (h)
250
200
150
100
50
0 0 10
300
20 30 40
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6. 50 students took a music examination and the cumulative frequency curve below
shows their marks (x), and the number of students who obtained less than x marks. The
highest possible mark is 120.
From the graph, estimate
(a) the number of students who scored less than 100 marks,
(b) the number of students who scored at least 60 but less than 100 marks,
(c) the value of b, such that 60% of the students scored less than b marks,
(d) what the pass mark should be, if the examination board wants 80% of students
to pass the examination.
Marks
120
50
40
30
20
10
0 20 40 60 80
Cu
mu
lati
ve
Fre
qu
ency
100
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7. The heights of 140 boys and 140 girls in a school were measured and recorded in the
following cumulative frequency curves.
(i) For both Boys and Girls, use the graphs to estimate
(a) the number of Girls who are less than or equal to 145 cm,
(b) the number of Boys who are taller than 160 cm,
(c) the number of students who are taller than 150 cm.
(ii) What is the gender of the tallest person in the school whose height was measured,
and what is his/her height?
Height (cm)
150
100
80
60
40
20
0 130 140
120
160 170
140
Cu
mu
lati
ve
Fre
qu
ency
180
Girls
Boys
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8. The scores of two classes in a recent Mathematics test were recorded and are displayed
in the following cumulative frequency curves. Students need at least 50 marks to pass
and at least 75 marks to get a distinction.
(i) For both Class A and B, use the graphs to estimate
(a) the number of students from Class A who passed the test,
(b) the total number of students that attained a distinction,
(ii) What was the highest mark attained, and which class was that student from?
(iii) A student is randomly selected. Find the probability that
(a) he/she is from Class A who attained a distinction.
(b) he/she is from Class B who failed the test.
(c) he/she got at least 60 but less than 70 marks for the test.
60
50
40
30
20
10
0 20 40 80 100
Cu
mu
lati
ve
Fre
qu
ency
Marks
Class A
Class B
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(iv) Later in the year, students from Class A took another test and the cumulative
frequency curve for the second test was plot against the results of their first test.
The curves are as shown below:
Which was a more difficult paper for students? Explain your answer.
60
50
40
30
20
10
0 20 40 80 100
Cu
mu
lati
ve
Fre
qu
ency
Marks
Test 1
Test 2
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9. The following cumulative frequency curve shows the average monthly income of 1
million Singaporeans from the year 2018.
(i) Using the graph
(a) estimate the percentage of the population who earn less than or equal to
$4000 a month,
(b) estimate the value of x, such that 50% of Singaporeans earned at least
$x or less.
(c) hence, calculate how much the highest earning Singaporean earns a
month as a percentage of $x.
Monthly Income ($)
12000
1
0.8
0.6
0.4
0.2
0 2000 4000 6000 8000
Cu
mu
lati
ve
Fre
qu
ency
(in
mil
lion
s)
10000 14000
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(ii) Using the data above, complete the frequency distribution table below.
Monthly Income Range ($x) Number of People (millions)
0 1000x
1000 1500x
1500 2000x
2000 2500x
2500 3000x
3000 4000x
4000 6000x
6000 13600x
(iii) Using the table, find an estimate of the mean Singaporean income.
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10. (i) A test with a maximum mark of 50 was given to a class of 36 students. The
cumulative frequency curve below shows the results (x marks) and the number
of students who obtained less than or equal to x marks.
On the same pair of axes, draw a cumulative frequency curve of the results (x
marks) and the number of students who obtained more than x marks for the same
test and the same class.
(ii) What does the intersection of the two curves tell us?
Cu
mu
lati
ve
Fre
qu
ency
Marks (x)
30
40
20
10
0 10 20
30
40 50
