Basic Engineering Mathematics, Fifth Edition

Presentation of statistical data 293

classesto form afrequency distribution. To assist in
accurately counting members in the various classes, a
tally diagramis used (see Problems 8 and 12).
Afrequency distributionis merely a table show-
ing classes and their corresponding frequencies (see
Problems 8 and 12). The new set of values obtained
by forming a frequency distribution is calledgrouped
data. The terms used in connection with grouped data
are shown in Figure 31.6(a). The size or range of a class
is given by theupper class boundary valueminus the
lower class boundary valueand in Figure 31.6(b) is
7. 65 − 7. 35 ;i.e., 0.30. Theclass intervalfor the class
shown in Figure 31.6(b) is 7.4 to 7.6 and the class
mid-point value is given by

(upper class boundary value)+(lower class boundary value)

2

and in Figure 31.6(b) is

7. 65 + 7. 35

2

i.e. 7. 5

(a) Class interval

(b)

Lower

class

boundary

Upper

class

boundary

Class

mid-point

7.35

to 7.3 7.4 to 7.6 7.7 to

7.5 7.65

Figure 31.6

One of the principal ways of presenting grouped data
diagrammatically is to use ahistogram,inwhichthe
areasof vertical, adjacent rectangles are made propor-
tional to frequencies of the classes (see Problem 9).
When class intervals are equal, the heights of the rect-
angles of a histogram are equal to the frequencies of the
classes. For histograms having unequal class intervals,
the area must be proportional to the frequency. Hence,
if the class interval of classAis twice the class inter-
val of class B, then for equal frequencies the height
of the rectangle representingAis half that ofB(see
Problem 11).
Anothermethod of presenting grouped data diagram-
matically is to use afrequency polygon, which is the
graph producedby plottingfrequency againstclass mid-
point values and joining the co-ordinates with straight
lines (see Problem 12).

A cumulative frequency distributionis a table

showing the cumulative frequency for each value of

upper class boundary. The cumulative frequency for a

particular value of upper class boundary is obtained by

adding the frequency of the class to the sum of the pre-

vious frequencies. A cumulative frequency distribution

is formed in Problem 13.

The curve obtained by joining the co-ordinates of

cumulative frequency (vertically) against upper class

boundary (horizontally) is called anogiveor acumu-

lative frequency distribution curve(see Problem 13).

Problem 8. The data given below refer to the gain

of each of a batch of 40 transistors, expressed

correct to the nearest whole number. Form a

frequency distribution for these data having seven

classes

81 83 87 74 76 89 82 84

86 76 77 71 86 85 87 88

84 81 80 81 73 89 82 79

81 79 78 80 85 77 84 78

83 79 80 83 82 79 80 77

Therangeof the data is the value obtained by tak-

ing the value of the smallest member from that of the

largest member. Inspection of the set of data shows that

range= 89 − 71 = 18 .The size of each class is given

approximately by the range divided by the number of

classes. Since 7 classes are required, the size of each

class is 18÷ 7 ;that is, approximately 3. To achieve

seven equal classes spanning a range of values from 71

to 89, the class intervals are selected as 70–72, 73–75,

and so on.

To assist with accurately determining the number in

each class, atally diagramis produced, as shown in

Table 31.1(a). This is obtained by listing the classes

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

40 members of the set in turn and allocating them to

the appropriate classes by putting ‘1’s in the appropri-

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

shown as an obliqueline crossing the four previous ‘1’s,

to help with final counting.

Afrequency distributionfor the data is shown in

Table 31.1(b) and lists classes and their correspond-

ing frequencies, obtained from the tally diagram. (Class

mid-point values are also shown in the table, since they

are used for constructing the histogram for these data

(see Problem 9).)