Statistical Methods in Medical Research 4ed

36 Describing data
107 01. The geometric mean, calculated from the logs and then transformed back
to the original scale, is 102 36. The median is 100 00, closer to the geometric
mean than to the arithmetic mean, as one might expect.
The log transform is one example of the way in which raw data can often
usefully be transformed to a new scale of measurement before analyses are
carried out. The use of transformations is described in more detail in §§10.8 and
11.10. In particular, the log transform will be seen to have other uses besides that
of reducing positive skewness.
2.6 Measures of variation
When the mean value of a series of measurements has been obtained, it is usually a
matter of considerable interest to express the degree of variation or scatter around
this mean. Are the readings all rather close to the mean or are some of them
scattered widely in each direction? This question is important for purely descriptive
reasons, as we shall emphasize below. It is important also since the measurement
of variation plays a central part in the methods of statistical inference which
are described in this book. To take a simple example, the reliability of the mean of
100 values of some variable depends on the extent to which the 100 readings differ
among themselves; if they show little variation the mean value is more reliable,
more precisely determined, than if the 100 readings vary widely. The role of
variation in statistical inference will be clarified in later chapters of this book.
At present we are concerned more with the descriptive aspects.
In works of reference it is common to find a single figure quoted for the value
of some biological quantity and the reader may not always realize that the stated
figure is some sort of average. In a textbook on nutrition, for example, we might
find the vitamin A content of Cheddar cheese given as 390 micrograms per 100
grams of cheese. Clearly, not all specimens of Cheddar cheese contain precisely
390 mg per 100 g; how much variation, then, is there from one piece of cheese to
another? To take another example from nutrition, the daily energy requirement
of a physically active man aged 25 years, of 180 cm and 73 kg, should be 12 0
megajoules. This requirement must vary from one person to another; how large
is the variation?
There is unlikely to be a single answer to questions of this sort, because the
amount of variation to be found in a series of measurements will usually depend
on the circumstances in which they are made and, in particular, on the way in
which these circumstances change from one reading to another. Specimens of
Cheddar cheese are likely to vary in their vitamin A content for a number of
reasons: major differences in the place and method of manufacture; variation in
composition from one specimen to another even within the same batch of
manufacture; the age of the cheese, and so on. Variation in the recorded
measurement may be partly due to measurement errorÐin the method of