Statistical Methods in Medical Research 4ed

42 Describing data
xi x will then need to be written with several significant digits and doubt will
arise as to whether an adequate number of significant digits was retained for x.
These difficulties have led to the widespread use of an alternative method of
calculating the sum of squares about the mean, P …xi x† 2 . It is based on the
fact that
P
… xi x†
2 ˆ P x 2 P
… xi†
i
2
: …2:3†
n
This is called the short-cut formula for the sum of squares about the mean.
The important point about (2.3) is that the computation is performed without
the need to calculate individual deviations from the mean, xi x. The sum of
squares of the original observations, xi, is corrected by subtraction of a quantity
dependent only on the mean (or, equivalently, the total) of the xi. This second
term is therefore often called the correction term, and the whole expression a
corrected sum of squares.
The previous example is reworked in Table 2.7. We again have the result
P
…xi x† 2 ˆ 100, and the subsequent calculations follow as in Table 2.6.
The short-cut formula avoids the need to square individual deviations with
many significant digits, but involves the squares of the xi, which may be large
numbers. This rarely causes trouble using a calculator, although care must be
taken to carry sufficient digits in the correction term to give the required number
of digits in the difference between the two terms. (For example, if P x2 i ˆ 2025
and … P xi† 2 =n ˆ 2019 3825, the retention of all these decimals will give
Table 2.7 Calculation of estimated variance and
standard deviation: short-cut formula (same data
as in Table 2.6).
xi
x 2
i
8 64
5 25
4 16
12 144
15 225
5 25
7 49
Ð Ð
56 548
P
…xi x† 2 ˆ P x2 i … P xi† 2 =n
ˆ 548 56 2 =7
ˆ 548 448
ˆ 100
Subsequent steps as in Table 2.6