Higher Engineering Mathematics

536 STATISTICS AND PROBABILITY

draw a histogram is to mark class mid-point values

on the horizontal scale and to draw the rectangles

symmetrically 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.

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.

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 shown
   in row 3, the cumulative frequency value is the sum
   of all frequencies having values of less than 7.95, i.e.
   3 + 5 + 9 =17, and so on. Theogivefor the cumu-
   lative frequency distribution given in Table 54.6 is
   shown in Fig. 54.11. The co-ordinates corresponding
   to each upper class boundary/cumulative frequency
   value are plotted and the co-ordinates are joined by
   straight lines (—not the best curve drawn through
   the co-ordinates as in experimental work.) The ogive
   is ‘anchored’ at its start by adding the co-ordinate
   (7.05, 0).

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

Figure 54.11

Now try the following exercise.

Exercise 207 Further problems on presen-

tation of grouped data

1. The mass in kilograms, correct to the nearest
   one-tenth of a kilogram, of 60 bars of metal
   are as shown. Form a frequency distribution
   of about 8 classes for these data.

39.8 40.3 40.6 40.0 39.6

39.6 40.2 40.3 40.4 39.8

40.2 40.3 39.9 39.9 40.0

40.1 40.0 40.1 40.1 40.2

39.7 40.4 39.9 40.1 39.9

39.5 40.0 39.8 39.5 39.9

40.1 40.0 39.7 40.4 39.3

40.7 39.9 40.2 39.9 40.0

40.1 39.7 40.5 40.5 39.9

40.8 40.0 40.2 40.0 39.9

39.8 39.7 39.5 40.1 40.2

40.6 40.1 39.7 40.2 40.3

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

There is no unique solution,

but one solution is:

39. 3 − 39 .41;39. 5 − 39 .65;

39. 7 − 39 .89;39. 9 − 40 .0 17;

40. 1 − 40 .2 15; 40. 3 − 40 .47;

40. 5 − 40 .64;40. 7 − 40. 82

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

2. Draw a histogram for the frequency distribu-
   tion given in the solution of Problem 1.
   ⎡

⎢

⎣

Rectangles, touching one another,

having mid-points of 39.35,

39 .55, 39.75, 39.95,...and

heights of 1, 5, 9, 17,...

⎤

⎥

⎦