Basic Engineering Mathematics, Fifth Edition

304 Basic Engineering Mathematics

intofourequalparts.Thus,fortheset{2,3,4,5,5,7,

9, 11, 13, 14, 17} there are 11 members and the values

of the members dividing the set into four equal parts

are 4, 7 and 13. These values are signified byQ 1 ,Q 2

andQ 3 and called the first, second and third quartile

values, respectively. It can be seen that the second quar-

tile value,Q 2 , is the value of the middle member and

hence is the median value of the set.

For grouped data the ogive may be used to determine

thequartilevalues.Inthiscase,pointsareselectedonthe

vertical cumulative frequency values of the ogive, such

that they divide the total value of cumulative frequency

into four equal parts. Horizontal lines are drawn from

these values to cut the ogive. The values of the variable

corresponding to these cutting points on the ogive give

the quartile values (see Problem 7).

When a set contains a large number of members, the

set can be split into ten parts, each containing an equal

number of members. These ten parts are then called

deciles.Forsetscontainingaverylargenumberofmem-

bers, the set may be split into one hundred parts,each

containing an equal number of members. One of these

partsiscalledapercentile.

Problem 7. The frequency distribution given

below refers to the overtime worked by a group of

craftsmen during each of 48 working weeks in a

year. Draw an ogive for these data and hence

determine the quartile values.

25–29 5

30–34 4

35–39 7

40–44 11

45–49 12

50–54 8

55–59 1

The cumulative frequency distribution (i.e. upper class

boundary/cumulative frequency values) is

29.5 5, 34.5 9, 39.5 16, 44.5 27,

49.5 39, 54.5 47, 59.5 48.

The ogive is formed by plottingthese values on a graph,

as shown in Figure 32.2. The total frequency is divided

into four equal parts, each having a range of 48÷4,

i.e. 12. This gives cumulative frequency values of 0 to

12 corresponding to the first quartile, 12 to 24 corre-

sponding to the second quartile, 24 to 36 corresponding

to the third quartile and 36 to 48 corresponding to the

fourth quartile of the distribution; i.e., the distribution

25

Cumulative frequency 10

40

30

20

50

30 35 Q 140 Q 245 Q 3

Upper class boundary values, hours

50 55 60

Figure 32.2

is divided into four equal parts. The quartile values

are those of the variable corresponding to cumulative

frequency values of 12, 24 and 36, markedQ 1 ,Q 2 and

Q 3 in Figure 32.2. These values, correct to the nearest

hour, are37 hours,43 hours and 48 hours, respec-

tively. TheQ 2 value is also equal to the median value

of the distribution. One measure of the dispersion of a

distribution is called thesemi-interquartile rangeand

is given by(Q 3 −Q 1 )÷2andis( 48 − 37 )÷2inthis

case; i.e., 5

1

2

hours.

Problem 8. Determine the numbers contained in

the (a) 41st to 50th percentile group and (b) 8th

decile group of the following set of numbers.

14 22 17 21 30 28 37 7 23 32

24 17 20 22 27 19 26 21 15 29

The set is ranked, giving

7141517171920212122

22 23 24 26 27 28 29 30 32 37

(a) Thereare20numbersintheset,hence thefirst 10%

will be the two numbers 7 and 14, the second 10%

willbe 15 and 17, and so on. Thus, the 41st to 50th

percentile group will be the numbers21 and 22.

(b) The first decile group is obtained by splitting the

ranked set into 10 equal groups and selecting the

first group; i.e., the numbers 7 and 14. The second

decile group is the numbers 15 and 17, and so on.

Thus, the 8th decile group contains the numbers

27 and 28.