Lesson notes on Cumulative frequency for students preparing for WASSCE, NECO and JAMB Exams

1. Cumulative Frequency
The cumulative frequency curve is drawn from a
cumulative frequency table.
We plot the upper class boundary against the cumulative
frequency

2. WASSCE MAY/JUNE 2018
𝑀𝑎𝑟𝑘𝑠 0 − 9 10 − 19 20 − 29 30 − 39 40 − 49 50 − 59 60 − 69 70 − 79 80 − 89 90 − 99
𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 10 28 40 92 𝑦 + 60 90 50 30 15 5
The table shows the distribution of marks scored by 500 candidates in an examination.
a) Find the value of 𝑦
b) Construct a cumulative frequency table and use it to draw a cumulative frequency curve.
c) Use the curve to estimate the probability of selecting a candidate who scored not more than 45%.
Solution
𝑇𝑜𝑡𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 = 500
10 + 28 + 40 + 92 + 𝑦 + 60 + 90 + 50 + 30 + 15 + 5
= 500
𝑦 = 80
Class
boundary
Upper class
boundary
Frequency Cumulative
Frequency
−0.5 − 9.5 9.5 10 10
9.5 − 19.5 19.5 28 38
19.5 − 29.5 29.5 40 78
29.5 − 39.5 39.5 92 170
39.5 − 49.5 49.5 140 310
49.5 − 59.5 59.5 90 400
59.5 − 69.5 69.5 50 450
69.5 − 79.5 79.5 30 480
79.5 − 89.5 89.5 15 495
89.5 − 99.5 99.5 5 500
Tracing from the graph, 250 students scored not more than 45%
P(Selecting a candidate who scored not more than 45%)=
250
500
=
1
2

3. JAMB 2001
The graph above shows the cumulative frequency of the distribution of masses of fertilizer for 48 workers in one institutio
Which of the following gives the interquartile range?
A. Q3 – Q1
B. Q3 – Q2
C. Q2 – Q1
D. ½ (Q3 – Q1)
Solution
The interquartile range is the third quartile minus the first quartile
The correct answer is A

4. JAMB 1984
The cumulative frequency function of the data below is given by the equation 𝑦 = 𝑐𝑓(𝑥). What is cf(5)?
A. 30
B. 32
C. 55
D. 62
E. 92
Solution
𝑐𝑓(5) Means adding up the score of 5
𝑐𝑓 5 = 30 + 32 + 30
= 92
The correct answer is E.
𝑺𝒄𝒐𝒓𝒆(𝒏) 𝑭𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚(𝒇)
3 30
4 32
5 30
6 35
8 20

5. JAMB 2004
The graph above shows the cumulative frequency curve of the distribution of marks in a class test. What percentage of th
students scored more than 20 marks?
A. 68%
B. 28%
C. 17%
D. 8%
Solution
𝑡ℎ𝑒 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑜𝑓 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 𝑤ℎ𝑜 𝑠𝑐𝑜𝑟𝑒𝑑 𝑚𝑜𝑟𝑒 𝑡ℎ𝑎𝑛
20 marks=
25−18
25
× 100%
=
7
25
× 100% = 28%

6. The mean of 2 − 𝑡, 4 + 𝑡, 3 − 2𝑡, 2 + 𝑡 and 𝑡 −
1 is.
JAMB 2018
SOLUTION
Mean, 𝑋 =
𝑥
𝑛
=
2 − 𝑡 + 4 + 𝑡 + 3 − 2𝑡 + 2 + 𝑡 + (𝑡 − 1)
5
2 + 4 + 3 + 2 − 1 − 𝑡 + 𝑡 − 2𝑡 + 𝑡 + 𝑡
5
10 + 0
5
10
2
= 2

7. The distribution of scores in a class test are as follows: 2,8,6,5,8,6,6,5,6 and 5. Find
the product of the modal and the median score
JAMB 2018
SOLUTION
6 occurs most
Modal mark= 6
For median mark, we arrange the numbers in order:
2,5,5,5,6,6,6,6,8,8
Median mark =
6+_6
2
12
2
= 6
Their product = 6 × 6 = 36

8. 1. The table below is a distribution of ages of pupils.
Which of the following points will be plotted when drawing a cumulative frequency
curve?
A. (4,2)
B. (3,5)
C. (8,10)
D. (7,10)
E. (0.5,2)
The correct answer is (8,10)
𝑴𝒂𝒓𝒌𝒔 (𝒙) 𝟏 − 𝟓 𝟔 − 𝟏𝟎 𝟏𝟏 − 𝟏𝟓 𝟏𝟔 − 𝟐𝟎 𝟐𝟏 − 𝟐𝟓
𝑁𝑜. 𝑜𝑓 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 2 8 5 3 4

9. 2. The table below is a distribution of ages of pupils.
Which of the following points will be the highest point on the cumulative frequency
curve?
A. (4,22)
B. (24,22)
C. (21,22)
D. (23,10)
E. (0.5,2)
The correct answer is (23,22)
𝑴𝒂𝒓𝒌𝒔 (𝒙) 𝟏 − 𝟓 𝟔 − 𝟏𝟎 𝟏𝟏 − 𝟏𝟓 𝟏𝟔 − 𝟐𝟎 𝟐𝟏 − 𝟐𝟓
𝑁𝑜. 𝑜𝑓 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 2 8 5 3 4

10. 3. The graph below is the cumulative frequency curve that represents marks scored in
a class test. Estimate the median mark from the graph.
A. 12.0
B. 13.0
C. 14.0
D. 15.0
E. 16.0
The correct answer is 12.0

11. 4. The graph below is the cumulative frequency curve that represents marks scored in
a class test. Estimate the third quartile from the graph.
A. 12.0
B. 13.0
C. 14.0
D. 15.0
E. 16.0
The correct answer is 16.0

12. 5. The graph below is the cumulative frequency curve that represents marks scored in
a class test. How many students sat for the test?.
A. 18
B. 19
C. 20
D. 21
E. 22
The correct answer is 22

13. 6. The graph below is the cumulative frequency curve that represents marks scored in a class test. What
is the probability of scoring a mark that is equal to the first quartile?.
A.
1
11
B.
2
11
C.
3
11
D.
4
11
E.
5
11
The correct answer is
4
11

14. 8. The table below is a distribution of ages of pupils.
Which of the following points will the cumulative frequency curve pass through?
A. (13,15)
B. (12,15)
C. (4,2)
D. (8,8)
E. (13,5)
The correct answer is (13,15)
𝑴𝒂𝒓𝒌𝒔 (𝒙) 𝟏 − 𝟓 𝟔 − 𝟏𝟎 𝟏𝟏 − 𝟏𝟓 𝟏𝟔 − 𝟐𝟎 𝟐𝟏 − 𝟐𝟓
𝑁𝑜. 𝑜𝑓 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 2 8 5 3 4

15. 9. The table below is a a cumulative frequency distribution table. What is the value of
𝑓?
A. 11.0
B. 12.0
C. 13.0
D. 14.0
E. 15.0
The correct answer is 12.0
Marks Frequency Cumulative frequency
1 − 4 3 𝑥
5 − 8 𝑦 7
9 − 12 2 9
13 − 16 𝑥 12

16. 10. The table below is a a cumulative frequency distribution table. What is the value of
𝑥 + 𝑦?
A. 8
B. 7
C. 9
D. 10
E. 11
The correct answer is 7
Marks Frequency Cumulative frequency
1 − 4 3 𝑥
5 − 8 𝑦 7
9 − 12 2 9
13 − 16 𝑥 12

17. Summary
1. Plot the marks against the cumulative frequencies
2. You can estimate the measures of spread from the
cumulative frequency curve
3. Trace
1
4
𝑓 𝑡ℎ value to estimate first quartile
4. Trace
1
2
𝑓 𝑡ℎ value to estimate first median
5. Trace
3
4
𝑓 𝑡ℎ value to estimate third quartile

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