Medians and quantiles

This is a section from my text book An Introduction to Medical Statistics, Fourth Edition. I hope that the topic will be useful in its own right, as well as giving a flavour of the book. Section references are to the book. 

4.5 Medians and quantiles

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  5.10
For 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 ith observation is the required quantile estimate. If not, let j be the integer part of i, the part before the decimal point. The quantile will lie between the jth and j + 1th observations. We estimate it by

x j + ( x j+1xj) × (ij)

For the median, for example, the 0.5 quantile, i = q ( n + 1) = 0.5 times (57+1) = 29, the 29th observation as before.

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 FEV

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). The following examples are for the FEV1 data and for serum triglyceride in cord blood for 282 babies:

Box and whisker plots for FEV, which is symmetrical, and triglyceride, which is highly skew

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 and whisker plots for four groups, side by side

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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