Introductory Biostatistics

1. It would be no use to take the mean of deviations because
   X
   ðx
   xÞ¼ 0


2. Taking the mean of the absolute values, for example
   P
   jx
   xj
   n

is a possibility. However, this measure has the drawback of being di‰cult

to handle mathematically, and we do not consider it further in this book.

The variances^2 (ssquared) is measured in the square of the units in which
thex’s are measured. For example, ifxis the time in seconds, the variance is
measured in seconds squared (sec^2 Þ. It is convenient, therefore, to have a mea-
sure of variation expressed in the same units as thex’s, and this can be done
easily by taking the square root of the variance. This quantity is thestandard
deviation, and its formula is

s¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

P

ðx
xÞ^2

n
 1

s

Consider again the data set

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

Calculation of the variances^2 and standard deviationsis illustrated in Table
2.9.
In general, this calculation process is likely to cause some trouble. If the
mean is not a ‘‘round’’ number, sayx¼ 10 =3, it will need to be rounded o¤,

TABLE 2.9

xx
x ðx
xÞ^2

80 0

5 
 39

4 
 416

12 4 16

15 7 49

5 
 39

P^7 
^11

x¼ 56

P

ðx
xÞ^2 ¼ 100

n¼ 7 s^2 ¼ 100 = 6 ¼ 16 : 67

x¼ 8 s¼

ffiffiffiffiffiffiffiffiffiffiffi

16 : 67

p

¼ 4 : 08

78 DESCRIPTIVE METHODS FOR CONTINUOUS DATA