and errors arise in the subtraction of this figure from eachx. This di‰culty can
easily be overcome by using the following shortcut formula for the variance:
s^2 ¼
P
x^2 ð
P
xÞ^2 =n
n 1
Our earlier example is reworked in Table 2.10, yielding identical results.
s^2 ¼
ð 548 Þð 56 Þ^2 = 7
6
¼ 16 : 67
When data are presented in the grouped form of a frequency table, the vari-
ance is calculated using the following modified shortcut formula:
s^2 F
P
fm^2 ð
P
fmÞ^2 =n
n 1
wherefdenotes an interval frequency,mthe interval midpoint calculated as in
Section 2.2.2 and the summation is across the intervals. This approximation is
illustrated in Table 2.11.
s^2 F
89 ; 724 : 25 ð 2086 : 5 Þ^2 = 57
56
¼ 238 : 35
sF 15 :4lb
(If individual weights were used, we would haves¼ 15 :9 lb.)
It is often not clear to beginners why we useðn 1 Þinstead ofnas the
denominator fors^2. This number,n1, called thedegrees of freedom, repre-
TABLE 2.10
xx^2
864
525
416
12 144
15 225
525
749
56 548
NUMERICAL METHODS 79