Higher Engineering Mathematics, Sixth Edition

Presentation ofstatistical data 539

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
cumulative frequency distribution given in Table 54.6
is shown in Fig. 54.11. The co-ordinates correspond-
ing to eachupper 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).

7.05

Cumulative frequency 10

40

30

20

50

7.35 7.65 7.95 8.25

Upper class boundary values in kilograms

8.55 8.85 9.15

Figure 54.11

Now try the following exercise

Exercise 206 Further problemson

presentation 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. 017 ;

40. 1 − 40. 215 ; 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 of1, 5 , 9 , 17 ,...

⎤

⎥

⎥

⎦

3. The information given below refers to the
   value of resistance in ohms of a batch of
   48 resistors of similar value. Form a fre-
   quencydistributionforthedata,havingabout6
   classes,anddrawa frequencypolygonandhis-
   togramtorepresent thesedatadiagramatically.

21.022.422.821.522.621.121.622.3

22.920.521.822.221.021.722.520.7

23.222.921.721.422.122.222.321.3

22.121.822.022.721.721.921.122.6

21.422.422.320.922.821.222.721.6

22.221.621.322.121.522.023.421.2

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

There is no unique solution,

but one solution is:

20.5–20.9 3;21.0–21.4 10;

21.5–21.9 11;22.0–22.4 13;

22.5–22.9 9;23.0–23.4 2

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

4. The time taken in hours to the failure of 50
   specimens of a metal subjected to fatigue fail-
   ure tests are as shown. Form a frequency dis-
   tribution, having about 8 classes and unequal
   class intervals, for these data.