Statistical Methods in Medical Research 4ed

2.6 Measures of variation 43
P
…xi x† 2 ˆ 5 6175; if the correction term had been rounded off to the nearest
whole number it would have given P …xi x† 2 ˆ 6Ðan accuracy of only 1
significant digit.) Indeed, this rounding error can cause problems in high-speed
computing, and computer programs should use the direct rather than the shortcut
formula for the sum of squares about the mean.
Most scientific calculators have a summation key which accumulates n, P x
and P x2 in stores and then the mean and standard deviation are each calculated
by successive keystrokes, the latter invoking calculation of (2.3). Calculators
usually have separate keys for (2.1a) and (2.2a), often called sn and sn 1,
respectively.
If the observations are presented in the form of a frequency distribution, and
the raw data are not retrievable, the standard deviation may be obtained, as
indicated earlier for the calculation of the mean, by assuming that all observations
are clustered at the mid-points of their grouping intervals. As with the
mean, the true standard deviation can then only be estimated, with some loss of
accuracy. There is an additional problem with the standard deviation and
variance, in that the calculation from grouped data tends to give somewhat too
high a value. An appropriate correction for this effect, called Sheppard's correction,
is to subtract 1
12 h2 from the calculated variance, h being the size of the
grouping interval. (If the grouping interval is not constant an average value may
be used.) The correction is rather small unless the grouping is quite crude, and
for this reason is often ignored. In the example based on the data of Table 2.4,
for which h ˆ 10, the standard deviation is estimated as 8 76 years, or 8 27 years
after applying Sheppard's correction.
The standard deviation of a set of measurements is expressed in the same
units as the measurements and hence in the same units as the mean. It is
occasionally useful to describe the variability by expressing the standard deviation
as a proportion, or a percentage, of the mean. The resulting measure, called
the coefficient of variation, is thus a dimensionless quantityÐa pure number. In
symbols,
CV…x† ˆ s
100% …2:4†
x
The coefficient of variation is most useful as a descriptive tool in situations
in which a change in the conditions under which measurements are made
alters the standard deviation in the same proportion as it alters the mean.
The coefficient of variation then remains unchanged and is a useful single
measure of variability. It is mentioned again in a more substantive context in
§5.3.
In §2.5 we described the geometric mean. By analogy, the geometric standard
deviation may be obtained by calculating the standard deviation of the logarithmically
transformed measurements and converting back to the original scale by