Basic Engineering Mathematics, Fifth Edition

296 Basic Engineering Mathematics

To assist with accurately determining the number

in each class, atally diagramis produced as shown

in Table 31.4. This is obtained by listing the classes in

the left-hand column and then inspecting each of the 50

members of the set of data in turn and allocating it to the

appropriateclass by puttinga ‘1’ in the appropriaterow.

Each fifth ‘1’ allocated to a particular row is marked as

an oblique line to help with final counting.

Table 31.4

Class Tally

7.1to7.3 111

7.4to7.6 1111

7.7to7.9 1111 1111

8.0to8.2 1111 1111 1111

8.3to8.5 1111 1111 1

8.6to8.8 1111 1

8.9to9.1 11

Afrequency distributionfor the data is shown in

Table 31.5 and lists classes and their corresponding fre-

quencies. Class mid-points are also shown in this table

since they are used when constructing the frequency

polygon and histogram.

Table 31.5

Class Class mid-point Frequency

7.1to7.3 7.2 3

7.4to7.6 7.5 5

7.7to7.9 7.8 9

8.0to8.2 8.1 14

8.3to8.5 8.4 11

8.6to8.8 8.7 6

8.9to9.1 9.0 2

A frequency polygonis shown in Figure 31.9,

the co-ordinates corresponding to the class mid-

point/frequency values given in Table 31.5. The co-

ordinates are joined by straight lines and the polygon

is ‘anchored-down’ at each end by joining to the next

class mid-point value and zero frequency.

Ahistogramis shown in Figure 31.10, the width

of a rectangle corresponding to (upper class boundary

7.2

4

2

0

6

Frequency

12

10

8

14

7.5 7.8 8.1

Class mid-point values

8.4 8.7 9.0

Frequency polygon

Figure 31.9

value – lower class boundary value) and height corre-

spondingtothe class frequency. The easiest way to draw

ahistogramistomarkclassmid-point valuesonthehori-

zontal scale and to draw the rectangles symmetrically

about the appropriate class mid-point values and touch-

ing one another. A histogram for the data given in

Table 31.5 is shown in Figure 31.10.

7.2

4

2

0

Frequency^6

10

8

12

14

7.5

7.35 7.65 7.95 8.25 8.55 8.85 9.15

7.8 8.1 8.4

Class mid-point values

8.7 9.0

Histogram

Figure 31.10

Problem 13. The frequency distribution for the

masses in kilograms of 50 ingots is

7.1 to 7.3 3

7.4 to 7.6 5

7.7 to 7.9 9

8.0 to 8.2 14

8.3 to 8.5 11

8.6 to 8.8 6

8.9 to 9.1 2

Form a cumulative frequency distribution for these

data and draw the corresponding ogive