Higher Engineering Mathematics, Sixth Edition

536 Higher Engineering Mathematics

Table 54.1(a)

Class Tally

70–72 1

73–75 11

76–78 1111 11

79–81 1111 1111 11

82–84 1111 1111

85–87 1111 1

88–90 111

Table 54.1(b)

Class Class mid-point Frequency

70–72 71 1

73–75 74 2

76–78 77 7

79–81 80 12

82–84 83 9

85–87 86 6

88–90 89 3

71

4

2

Frequency^6

10

8

12

14

16

74 77 80 83

Class mid-point values

8986

Figure 54.7

Problem 10. The amount of money earned

weekly by 40 people working part-time in a factory,

correct to the nearest £10, is shown below. Form a

frequency distribution having 6 classes for these

data.

80 90 70 110 90 160 110 80

140 30 90 50 100 110 60 100

80 90 110 80 100 90 120 70

130 170 80 120 100 110 40 110

50 100 110 90 100 70 110 80

Inspection of the set given shows that the major-

ity of the members of the set lie between £80

and £110 and that there are a much smaller num-

ber of extreme values ranging from £30 to £170.

If equal class intervals are selected, the frequency

distribution obtained does not give as much infor-

mation as one with unequal class intervals. Since

the majority of members are between £80 and £100,

the class intervals in this range are selected to be

smaller than those outside of this range. There is no

unique solution and one possible solution is shown in

Table 54.2.

Table 54.2

Class Frequency

20–40 2

50–70 6

80–90 12

100–110 14

120–140 4

150–170 2

Problem 11. Draw a histogram for the data given

in Table 54.2.

When dealing with unequal class intervals, the his-

togram must be drawn so that the areas, (and not the

heights), of the rectangles are proportional to the fre-

quencies of the classes. The data given are shown in

columns 1 and 2 of Table 54.3. Columns 3 and 4 give

the upper and lower class boundaries, respectively. In

column 5, the class ranges (i.e. upper class bound-

ary minus lower class boundary values) are listed. The

heights of the rectangles are proportional to the ratio

frequency

class range

, as shown in column 6. The histogram is

shown in Fig. 54.8.