Statistical Methods in Medical Research 4ed

40 Describing data
Therefore the mean of the deviations xi x will always be zero. We could, however,
take the mean of the deviations ignoring their sign, i.e. counting them all as
positive. These quantities are called the absolute values of the deviations and are
denoted by jxi xj. Their mean, P jxi xj=n, is called the mean deviation. This
measure has the drawback of being difficult to handle mathematically, and we
shall not consider it any further in this book.
Another way of getting over the difficulty caused by the positive and negative
signs is to square them. The mean value of the squared deviations is called the
variance and is a most important measure in statistics. Its formula is
P
…xi x†
Variance ˆ
2
: …2:1†
n
The numerator is often called the sum of squares about the mean. The variance is
measured in the square of the units in which x is measured. For example, if x is
height in cm, the variance will be measured in cm2 . It is convenient to have a
measure of variation expressed in the original units of x, and this can be easily
done by taking the square root of the variance. This quantity is known as the
standard deviation, and its formula is
P
…xi x†
Standard deviation ˆ
2
s"
#
: …2:1a†
n
In practice, in calculating variances and standard deviations, the n in the
denominator is almost always replaced by n 1. The reason for this is that in
applying the methods of statistical inference, developed later in this book, it is
useful to regard the collection of observations as being a sample drawn from a
much larger group of possible readings. The large group is often called a
population. When we calculate a variance or a standard deviation we may wish
not merely to describe the variation in the sample with which we are dealing, but
also to estimate as best we can the variation in the population from which the
sample is supposed to have been drawn. In a certain respect (see §5.1) a better
estimate of the population variance is obtained by using a divisor n 1 instead of
n. Thus, we shall almost always use the formula for the estimated variance or
sample variance:
Estimated variance, s 2 P
…xi x†
ˆ
2
, …2:2†
n 1
and, similarly,
P
…xi x†
Estimated standard deviation, s ˆ
2
s"
#
: …2:2a†
n 1
Having established the convention we shall very often omit the word `estimated'
and refer to s 2 and s as `variance' and `standard deviation', respectively.