Page 84 - Becoming Metric Wise

74 Becoming Metric-Wise

Figure 4.5 Data shown in Table 4.1 represented as a frequency polygon.
4.3.1 The Arithmetic Average

The arithmetic average or mean, of a sequence of values (x i ) i51,...,n is
denoted x and defined as:
n
1 X
x 5 x i (4.1)
n
i51
The famous Journal Impact Factor, discussed in Chapter 6, Journal
Citation Analysis, is actually an average number of received citations.
If data are only given by a frequency table and m j is the mid-point of the
j-th class, then the average is obtained—approximately—by:
k
1 X
x 5 f j m j (4.2)
n
j51
where k is the number of classes and f j is the number of elements in the
j-th class.

4.3.2 The Geometric and the Harmonic Mean

Given a sequence of strictly positive values (y i ) i51,.. .,n then its geometric
mean is defined as
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n (4.3)
y 1 :y 2 .. .y n
Its harmonic mean is:
n
P n 1 (4.4)
j51 y j