Statistical Methods in Medical Research 4ed

38 Describing data
If the number of observations is not too small, a modification may be
introduced which avoids the use of the absolute extreme values. If the readings
are arranged in ascending or descending order, two values may be ascertained
which cut off a small fraction of the observations at each end, just as the median
breaks the distribution into two equal parts. The value below which a quarter of
the observations fall is called the lower quartile, that which is exceeded by a
quarter of the observations is called the upper quartile, and the distance between
them is called the interquartile range. This measure is not subject to the second
disadvantage of the range and is less subject to the other disadvantages.
The evaluation of the quartiles for a set of n values is achieved by first
calculating the corresponding ranks by
and
rl ˆ 1 1
4 n ‡ 2
ru ˆ 3 1
4 n ‡ 2
and then calculating the quartiles as the corresponding values in the orderd set of
values, using interpolation if necessary. For example, in the data on duration of
absence from work (p. 32) where n is 21, rl ˆ 5 3
4 and ru ˆ 16 1
4 . The lower quartile
is then obtained by interpolation between the 5th and 6th values; these are both 3
days, so the lower quartile is also 3 days. The upper quartile is obtained by
interpolation between the 16th and 17th values, 7 and 8 days. The interpolation
involves moving 1 4 of the way from the 16th value towards the 17th value, to
give 7 ‡ 1
1
4 …8 7† ˆ7 4 days. It should be noted that there is not a single
standard convention for calculating the quartiles. Some authors define the
quartiles in terms of the ranks rl ˆ 1
4 …n ‡ 1† and ru ˆ 3
4 …n ‡ 1†. It is also common
to round 1 3
4 and 4 to the nearest integer and to use interpolation only when this
involves calculation of the mid-point between two values. Differences between
the results using the different conventions are usually small and unimportant in
practice.
The quartiles are particular examples of a more general index, the percentile
(or centile). The value below which P% of the values fall is called the Pth
percentile. Thus, the lower and upper quartiles are the 25th and 75th percentiles,
respectively. The quartiles or any other percentiles could be read off a plot of
cumulative relative frequency such as Fig. 2.9. The term quantile is used when the
proportion is expressed as a fraction rather than a percentage: thus, the 30th
percentile is the same as the 0 3 quantile.
A convenient method of displaying the location and variability of a set of
data is the box-and-whisker plot. The basic form of this plot shows a box defined
by the lower and upper quartiles and with the median marked by a subdivision of
the box. The whiskers extend from both ends of the box to the minimum and