We often want to summarize a frequency distribution in a few numbers, for ease of reporting or comparison. The most direct method is to use quantiles. The quantiles are values which divide the distribution such that there is a given proportion of observations below the quantile. For example, the median is a quantile. The median is the central value of the distribution, such that half the observations are less than or equal to it and half are greater than or equal to it. We can estimate any quantiles easily from the cumulative frequency distribution or a stem and leaf plot. For example, the following data are measurements of Forced Expiratory Volume in one second (FEV1) for 57 male medical students:
2.85 3.19 3.50 3.69 3.90 4.14 4.32 4.50 4.80 5.20 2.85 3.20 3.54 3.70 3.96 4.16 4.44 4.56 4.80 5.30 2.98 3.30 3.54 3.70 4.05 4.20 4.47 4.68 4.90 5.43 3.04 3.39 3.57 3.75 4.08 4.20 4.47 4.70 5.00 3.10 3.42 3.60 3.78 4.10 4.30 4.47 4.71 5.10 3.10 3.48 3.60 3.83 4.14 4.30 4.50 4.78 5.10For the FEV1 data the median is 4.1, the 29th value in Table 4.4. If we have an even number of points, we choose a value midway between the two central values.
In general, we estimate the q quantile, the value such that a
proportion q will be below it, as follows. We have n ordered
observations which divide the scale into n + 1 parts: below the
lowest observation, above the highest and between each adjacent pair. The
proportion of the distribution which lies below the i th observation
is estimated by i / (n + 1). We set this equal to q and
get i = q( n + 1). If i is an integer, the
observation is the required quantile estimate. If not, let
the integer part of i, the part before the decimal point. The quantile
will lie between the jth and j + 1th observations. We estimate
Other quantiles which are particularly useful are the quartiles of the distribution. The quartiles divide the distribution into four equal parts, called fourths or quarters. The second quartile is the median. For the FEV1 data the first and third quartiles are 3.54 and 4.53. For the first quartile, i = 0.25 times 58 = 14.5. The quartile is between the 14th and 15th observations, which are both 3.54. For the third quartile, i=0.75 times 58 = 43.5, so the quartile lies between the 43rd and 44th observations, which are 4.50 and 4.56. The quantile is given by 4.50 + (4.56 − 4.50) × (43.5 − 43) = 4.53. We often divide the distribution at 99 centiles or percentiles. The median is thus the 50th centile. For the 20th centile of FEV1, i = 0.2 × 58 = 11.6, so the quantile is between the 11th and 12th observations, 3.42 and 3.48, and can be estimated by 3.42 + (3.48 - 3.42) × (11.6 − 11) = 3.46.
We can also estimate these easily from the cumulative frequency polygon
Figure 4.2 Cumulative frequency polygon of FEV1 (data
from Physiology practical class, St George’s Hospital Medical
Figure 4.2 Cumulative frequency polygon of FEV1 (data from Physiology practical class, St George’s Hospital Medical School).
We find the position of the quantile on the vertical axis, e.g. 0.2 for the 20th centile or 0.5 for the median, draw a horizontal line to intersect the cumulative frequency polygon, and read the quantile off the horizontal axis. The term ‘quartile’ is often used incorrectly to mean the fourth or quarter of the observations which fall between two quartiles. The related words ‘quintile’ and ‘tertile’ often suffer in the same way.
Tukey (1977) used the median, quartiles, maximum and minimum as a convenient
five figure summary of a distribution. He also suggested a neat graph,
the box and whisker plot , which represents this (Figure 4.16).
examples are for the FEV1 data and for serum triglyceride in cord blood
for 282 babies:
Figure 4.16 Box and whisker plots for FEV1 and for serum
triglyceride (data from Physiology practical class, St George’s
Hospital Medical School/Tessi Hanid).
Figure 4.16 Box and whisker plots for FEV1 and for serum triglyceride (data from Physiology practical class, St George’s Hospital Medical School/Tessi Hanid).
The examples in Figure 4.16 are for the FEV1 data and for serum
triglyceride in cord blood for 282 babies.
The box shows the distance between the quartiles, with the median marked
as a line, and the ‘whiskers’ show the extremes. The different shapes of
the FEV1 and serum triglyceride distributions is clear from the graph.
The different shapes of the FEV1 and serum
triglyceride distributions are clear from the graph.
For display purposes, an observation whose distance from the edge of
the box (i.e. the quartile) is more than 1.5 times the length of the box
(i.e. the interquartile range, Section 4.7) may be called an outlier.
Outliers may be shown as separate points.
The plot is useful for showing the comparison of several groups
Figure 4.17 Box plots showing a roughly symmetrical
variable in four groups, with an outlying point (data in
Table 10.7) (data supplied by Moses Kapembwa, personal
Figure 4.17 Box plots showing a roughly symmetrical variable in four groups, with an outlying point (data in Table 10.7) (data supplied by Moses Kapembwa, personal communication).
This example shows a fat absorbtion test in patients who have AIDS, AIDS Related complex, are HIV positive but asymptomatic, and normal controls.
Adapted from pages 49–50 of An Introduction to Medical Statistics by Martin Bland, 2015, reproduced by permission of Oxford University Press.
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