Introductory Biostatistics

skewed distributions, analyses are commonly done on the log scale. After
obtaining a mean on the log scale, we should take the antilog to return to the
original scale of measurement; the result is called thegeometric meanof thex’s.
The e¤ect of this process is to minimize the influences of extreme observations
(very large numbers in the data set). For example, considering the data set

f 8 ; 5 ; 4 ; 12 ; 15 ; 7 ; 28 g

with one unusually large measurement, we have Table 2.8, with natural logs
presented in the second column. The mean is

x¼

79

7

¼ 11 : 3

while on the log scale we have

P

lnx

n

¼

15 : 55

7

¼ 2 : 22

leading to a geometric mean of 9.22, which is less a¤ected by the large mea-
surements. Geometric mean is used extensively in microbiological and sero-
logical research, in which distributions are often skewed positively.

Example 2.7 In some studies the important number is the time to an event,
such as death; it is called thesurvival time. The termsurvival timeis conven-
tional even though the primary event could be nonfatal, such as a relapse or the
appearance of the first disease symptom. Similar to cases of income and anti-
body level, the distributions of survival times are positively skewed; therefore,
data are often summarized using the median or geometric mean. The following
is a typical example.

TABLE 2.8

x lnx

8 2.08

5 1.61

4 1.39

12 2.48

15 2.71

7 1.95

28 3.33

79 15.55

NUMERICAL METHODS 75