MEASURES OF CENTRAL TENDENCY AND DISPERSION 543J
heights of the 100 people given in Problem 1
of Exercise 209, page 540. [9.394 cm]- Calculate the standard deviation from the
mean for the data given in Problem 3 of Exer-
cise 209, page 541, correct to 3 significant
figures. [0.00544 cm]
55.5 Quartiles, deciles and percentiles
Other measures of dispersion which are sometimes
used are the quartile, decile and percentile values.
Thequartile valuesof a set of discrete data are
obtained by selecting the values of members which
divide the set into four equal parts. Thus for the set:
{2, 3, 4, 5, 5, 7, 9, 11, 13, 14, 17}there are 11 mem-
bers 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, sec-
ond and third quartile values, respectively. It can be
seen that the second quartile 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 deter-
mine the quartile values. In this case, points are
selected on the vertical cumulative frequency val-
ues 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 calleddeciles. For sets containing a very large
number of members, the set may be split into one
hundred parts, each containing an equal number of
members. One of these parts is called apercentile.
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.
25–29 5, 30–34 4, 35–39 7,
40–44 11, 45–49 12, 50–54 8,
55–59 1Draw an ogive for this data and hence determine
the quartile values.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 48The ogive is formed by plotting these values on a
graph, as shown in Fig. 55.2. The total frequency is
divided into four equal parts, each having a range of
48/4, i.e. 12. This gives cumulative frequency val-
ues of 0 to 12 corresponding to the first quartile,
12 to 24 corresponding to the second quartile, 24 to
36 corresponding to the third quartile and 36 to 48
corresponding to the fourth quartile of the distribu-
tion, i.e. the distribution 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 andQ 3 in Fig. 55.2. These
values, correct to the nearest hour, are37 hours,
43 hours and 48 hours, respectively. 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 )/2, and is (48−37)/2 in this case, i.e.
512 hours.Figure 55.2Problem 8. Determine the numbers contained
in the (a) 41st to 50th percentile group, and
(b) 8th decile group of the set of numbers shown
below:
14 22 17 21 30 28 37 7 23 32
24 17 20 22 27 19 26 21 15 29