Marks
10
20
30
40
50
60
70
80
90
Frequency
1
1
x
5
y
1
4
3
1
The frequency distribution shows the marks distribution of a class of 30 students in an examination.
The mean mark of the distribution is 52.
(a) Find the values of x and y.
(b) Construct a group frequency distribution table starting with a lower class limit of 1 and class interval of 10.
(c) Draw a histogram for the distribution
(d) Use the histogram to estimate the mode.
Verified Correct Answer: .
Step-by-Step Mathematical Solution
(a)
Marks(x)
Frequency (f)
fx
10
1
10
20
1
20
30
x
30x
40
5
200
50
y
50y
60
1
60
70
4
280
80
3
240
90
1
90
Total
16 + x + y
900 + 30x + 50y
\(\sum f = 16 + x + y = 30\)
\(\implies x + y = 14 ... (1)\)
\(\bar{x} = \frac{\sum fx}{\sum f}\)
\(52 = \frac{900 + 30x + 50y}{30}\)
\(1560 = 900 + 30x + 50y \implies 30x + 50y = 660\)
\(3x + 5y = 66 .... (2)\)
Solving equation (1) and (2),
From (1), x = 14 - y
\(3(14 - y) + 5y = 42 - 3y + 5y = 66\)
\(2y = 24 \implies y = 12\)
\(x = 14 - y = 14 - 12 = 2\)
(x, y) = (2, 12).
(b)
Class interval
Frequency
Upper class boundary
1 - 10
1
10.5
11 - 20
1
20.5
21 - 30
2
30.5
31 - 40
5
40.5
41 - 50
12
50.5
51 - 60
1
60.5
61 - 70
4
70.5
71 - 80
3
80.5
81 - 90
1
90.5
(c)
(d) From the histogram, Mode = 44.
Source: Myschool Past Questions Archive (Record #52323)