Principles and Practice of Pharmaceutical Medicine

Group B of those who do not. The results are
presented in Table 25.4.
We notice that the two groups of subjects have
different means and different standard deviations.
Both standard deviations are smaller than the one
obtained before separating the subjects into sub-
groups. That is, the two groups are more homo-
geneous than the original group. When one
combines the standard deviations into so-called
pooled standard deviation, the result is
S.D.pooled¼2.47 which is substantially lower
than the standard deviation of the original com-
bined group. The reason for this is that when we
calculated the mean of the combined group, we
ignored the fact that the group consisted of two
subgroups with different means. Thus, the calcu-
lated mean was in fact, a mean of the two sub-
groups’ means. Indeed, the overall mean 56.8
equals the average of the means of the two sub-
groups, that is, 56: 8 ¼ð 54 : 2 þ 59 : 4 Þ=2. The stan-
dard deviation, therefore, represented the sum of
two sources of variation: the intra-group variability

represented by the two subgroups’ standard

deviations, and the intergroup variability repre-

sented by the difference between the two sub-

groups’ means. In general, if one combines two

groups of measurements with the same number of

measurements in each group and if the standard

deviations of the two groups and the combined

group are denoted byS 1 ,S 2 andS, respectively,

and if the means of the two groups are denoted by

M 1 andM 2 , respectively, then the following rela-

tionship holds:

S^2 ¼

1

2

ðS^21 þS^22 Þþ

ðM 1 
M 2 Þ

2

2

The variance of the combined group is the sum of

the two parts: The first is the average of the var-

iances of the two individual groups, and the second

is one half of the square of the difference between

the means of the two groups. The first part repre-

sents the intra-group variation and the second the

intergroup variation.

Table 25.3 Heartbeat

measurements of 20 students

Student Heartbeat

160

253

356

456

556

657

756

852

963

10 51

11 59

12 63

13 55

14 58

15 56

16 53

17 64

18 56

18 58

20 55

Mean 56.8

S.D. 3.57

Table 25.4 Heartbeat measurements of 20 students

by exercise status

Subject Heartbeat

Group A 2 53

852

756

13 55

20 55

556

455

18 56

10 51

16 53

Mean 54.2

S.D. 1.81

Group B 12 63

15 56

14 58

160

11 59

657

963

356

17 64

19 58

Mean 59.4

S.D. 2.99

25.6 THE CONTROLLED CLINICAL TRIAL: BASIC DESIGN ELEMENTS 321