basic_engineering_mathematics0

Measures of central tendency and dispersion 239
and standard deviation,
√ { ∑{f } (x −¯x)2 }
σ = ∑ = √ 0.41569
f
= 0.645, correct to 3 significant figures
Now try the following exercise
Exercise 114
Further problems on standard deviation
(Answers on page 283)
1. Determine the standard deviation from the mean of the
set of numbers:
{35, 22, 25, 23, 28, 33, 30}
correct to 3 significant figures.
2. The values of capacitances, in microfarads, of ten capacitors
selected at random from a large batch of similar
capacitors are:
34.3, 25.0, 30.4, 34.6, 29.6, 28.7,
33.4, 32.7, 29.0 and 31.3
Determine the standard deviation from the mean for
these capacitors, correct to 3 significant figures.
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.
4. Calculate the standard deviation from the mean for the
mass of the 50 bricks given in Problem 1 of Exercise
113, page 237, correct to 3 significant figures.
5. Determine the standard deviation from the mean, correct
to 4 significant figures, for the heights of the 100 people
given in Problem 2 of Exercise 113, page 237.
6. Calculate the standard deviation from the mean for the
data given in Problem 3 of Exercise 113, page 237,
correct to 3 decimal places.
31.5 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. 29.2. 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. 31.2. 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