Higher Engineering Mathematics, Sixth Edition

538 Higher Engineering Mathematics

to the appropriate class by putting a ‘1’ in the appropri-

ate row. Each fifth ‘1’ allocated to a particular row is

marked as an oblique line to help with final counting.

Afrequency distributionfor the data is shown in

Table 54.5 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 54.5

Class Class mid-point Frequency

7.1to7.3 7.2 3

7.4to7.6 7.5 5

7.5to7.9 7.8 9

8.0to8.2 8.1 14

8.1to8.5 8.4 11

8.2to8.8 8.7 6

8.9to9.1 9.0 2

A frequency polygon is shown in Fig. 54.9, the

co-ordinates corresponding to the class mid-

point/frequency values, given in Table 54.5. The

co-ordinates are joined by straight lines and the poly-

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

next class mid-point value and zero frequency.

7.2

4

2

0

6

Frequency

12

10

8

14

7.87.5 8.1

Class mid-point values

8.4 8.7 9.0

Frequency polygon

Figure 54.9

Ahistogramis shown in Fig. 54.10, the width of

a rectangle corresponding to (upper class boundary

value—lower class boundary value) and height corre-

spondingtothe class frequency. The easiest way to draw

a histogram is to mark class mid-point values on the

horizontal scale and to draw the rectangles symmetri-

cally about the appropriate class mid-point values and

touching one another. A histogram for the data given in

Table 54.5 is shown in Fig. 54.10.

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

9.08.7

Histogram

Figure 54.10

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,

8.9to9.1 2,

Form a cumulative frequency distribution for these

data and draw the corresponding ogive.

Acumulative frequency distributionis a table giv-

ing values of cumulative frequency for the value of

upper class boundaries, and is shown in Table 54.6.

Columns 1 and 2 show the classes and their frequen-

cies. Column 3 lists the upper class boundary values

for the classes given in column 1. Column 4 gives

the cumulative frequency values for all frequencies

less than the upper class boundary values given in

column 3. Thus, for example, for the 7.7 to 7.9 class

Table 54.6

1 2 3 4

Class Frequency Upper Class Cumulative

boundary frequency

Less than

7.1–7.3 3 7.35 3

7.4–7.6 5 7.65 8

7.7–7.9 9 7.95 17

8.0–8.2 14 8.25 31

8.3–8.5 11 8.55 42

8.6–8.8 6 8.85 48

8.9–9.1 2 9.15 50