we are now able to define a number of important summarized measures, start-
ing with the arithmetic average ormean.
2.2.1 Mean
Given a data set of sizen,
fx 1 ;x 2 ;...;xng
the mean of thex’s will be denoted byx(‘‘x-bar’’) and is computed by sum-
ming all thex’s and dividing the sum byn. Symbolically,
x¼
P
x
n
It is important to know thatS(‘‘sigma’’) stands for an operation (that of
obtaining the sum of the quantities that follow) rather than a quantity itself.
For example, considering the data set
f 8 ; 5 ; 4 ; 12 ; 15 ; 5 ; 7 g
we have
n¼ 7
X
x¼ 56
leading to
x¼
56
7
¼ 8
Occasionally, data, especially secondhand data, are presented in the grouped
form of a frequency table. In these cases, the meanxcan be approximated
using the formula
xF
P
ðfmÞ
n
wherefdenotes the frequency (i.e., the number of observations in an interval),
mthe interval midpoint, and the summation is across the intervals. The mid-
point for an interval is obtained by calculating the average of the interval lower
true boundary and the upper true boundary. For example, if the first three
NUMERICAL METHODS 73