546 Higher Engineering Mathematics
3. The tensile strength in megapascals for 15
samples of tin were determined and found
to be:
34.61, 34.57, 34.40, 34.63, 34.63,
34.51, 34.49, 34.61, 34.52, 34.55,
34.58, 34.53, 34.44, 34.48 and 34.40
Calculate the mean and standard deviation
from the mean for these 15 values, correct to
4 significant figures.
mean 34.53 MPa, standard
deviation 0.07474 MPa
4. Determine the standard deviation from the
mean, correct to 4 significant figures, for the
heights of the 100 people given in Problem 1
of Exercise 208, page 544.
[9.394 cm]
5. Calculate the standard deviation from the
mean for the data given in Problem 3 of
Exercise 208, page 544, 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.
The quartile values of 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 members
and the values of the members dividing the set into four
equal parts are 4, 7, and 13. These values are signified by Q 1 , Q 2 and Q 3 and called the first, second 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 determine the
quartile values. In this case, points are selected on the
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. 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 a percentile.
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 1
Draw 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 48
The 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 values 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 distribution, 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, marked Q 1 , Q 2 and Q 3 in Fig. 55.2.
These values, correct to the nearest hour, are 37 hours,
43 hours and 48 hours, respectively. The Q 2 value is
also equal to the median value of the distribution. One
measure of the dispersion of a distribution is called the
semi-interquartile range and is given by (Q 3 − Q 1 )/2,
and is (48 −37)/2 in this 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 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
3. The tensile strength in megapascals for 15
samples of tin were determined and found
to be:
34.61, 34.57, 34.40, 34.63, 34.63,
34.51, 34.49, 34.61, 34.52, 34.55,
34.58, 34.53, 34.44, 34.48 and 34.40
Calculate the mean and standard deviation
from the mean for these 15 values, correct to
4 significant figures.
mean 34.53 MPa, standard
deviation 0.07474 MPa
4. Determine the standard deviation from the
mean, correct to 4 significant figures, for the
heights of the 100 people given in Problem 1
of Exercise 208, page 544.
[9.394 cm]
5. Calculate the standard deviation from the
mean for the data given in Problem 3 of
Exercise 208, page 544, 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.
The quartile values of 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 members
and the values of the members dividing the set into four
equal parts are 4, 7, and 13. These values are signified by Q 1 , Q 2 and Q 3 and called the first, second 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 determine the
quartile values. In this case, points are selected on the
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. 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 a percentile.
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 1
Draw 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 48
The 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 values 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 distribution, 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, marked Q 1 , Q 2 and Q 3 in Fig. 55.2.
These values, correct to the nearest hour, are 37 hours,
43 hours and 48 hours, respectively. The Q 2 value is
also equal to the median value of the distribution. One
measure of the dispersion of a distribution is called the
semi-interquartile range and is given by (Q 3 − Q 1 )/2,
and is (48 −37)/2 in this 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 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