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