Median (calculation)

Table of Contents

The median is a statistical technology used as a measure of central tendency when doing analyses at ordinal levels. It is a technology in two accounts: it is derived by using a particular procedure and, consequently, refers to the resulting value. Indeed, the median is that value which divides a ranked distribution of scores exactly in half, so that 50% of the distribution is below the median and 50% of it is above the median.

Interpreting a median

There seems to be no standard notation for the median. In any case, the median is interpreted as the "middle value" in a distribution of scores, or as the "average value" of an ordinal variable. A more formal definition is the following: the median is that value at the 50^th percentile in a distribution of scores, and it splits such distribution in half (thus, half the scores are above and half the scores are below the median)⁴.

For example, in the following ranked sequence,

1, 2, 3, 4, 5

the median is '3', as it is the middle value in the sequence. When there are odds cases in the distribution, the median identifies a 'real' case as the average. This median divides the sample approximately in half. In this next sequence, however,

1, 2, 4, 5, 8, 15

the median falls between 4 & 5, the two middle values. When there are even cases in the distribution, the median always falls between two cases and it is, thus, calculated as the mean of both, ie 4.5 [(4+5)/2]. Such median does not identify a real case but it is able to divide the sample in exactly two equal halves.

Following from above examples, it is clear that a median represents the middle score in a ranked distribution and it is not affected by the actual value of particular scores (eg, '8' and '15' are, simply, next in the sequence, independently of their actual value). This property makes the median both an average not influenced by extreme cases and a good indicator to know something of, at least, half the sample.

For example, imaging that you are told that a group of schoolchildren were given a test of happiness, which measured happiness on a scale ranging from 1, 'Very unhappy' to 5, 'Very happy'. You are also told that the median for the group was 4, 'Happy'.

Even without any other information, the median literally informs that 4, 'Happy' is the ordinal average of the distribution of scores but also that at least 50% of the scores are either 4, 'Happy' or 5, 'Very happy'. Of course, another way of looking at it is that 50% of the scores may be 1, 'Very unhappy', 2, 'Unhappy', 3, 'Neither happy nor unhappy', and, even, 4, 'Happy'. But whatever the point-of-view chosen, the median is still closer to the upper end of the happiness scale. Thus, assuming that each child contributed one score only, we can infer that these schoolchildren are overall happy, with at least half the class reporting being happy or very happy.

Properties of the median

The median is a statistical technology for identifying the middle value in a distribution. It is based on a procedure (identifying the mid-value in an ordinal distribution), not on a mathematical formula³.
The median is a descriptive statistic which provides that value which splits a distribution of scores right in the middle in regards to a particular variable⁴.
The median can be used as a measure of central tendency. The median is particularly useful as the non-parametric equivalent to the mean, when variables are not normally distributed or there are outliers in the distribution. In fact, the median should be the preferred central tendency statistic for describing averages when using an ordinal level of analysis (eg, with ordinal variables and with non-normally distributed interval and ratio variables).
When samples are normally distributed, the median equals both the mode and the mean. Otherwise, it may differ from those two measures, sometimes greatly.
The median can be used with ordinal, interval and ratio variables.
A good measure of dispersion around the median is the interquartile range and the standard percentile range.
A good graphic for the median is the boxplot.

References

1. LUND RESEARCH (2007). Measures of central tendency. Retrieved from Stats4Students.com on 15 May 2010.

+++ Footnotes +++

2. In certain cases, the formula for the mean may be used for calculating the median. All-in-all, however, obtaining the median requires a procedure, not a mathematical formulation.

3. A formula is often provided not for calculating the median but for finding it by counting values from either extreme of the range up to the count given by the formula. The formula is: median count = (range + 1) / 2.

4. When the scores are independent, thus coming from different individuals, we could also say that the median actually divides the group of individuals (eg, the sample or the population) in, approximately, half. This is not the case if scores are not independent, of course.

Want to know more?

LUND RESEARCH (2007). Measures of central tendency. Retrieved from Stats4Students.com on 15 May 2010.

Wikipedia - Median: You can learn a bit more about the median in this page in Wikipedia.
Khan Academy - Mean, median and mode: Khan teaches about central tendency measures (mean, median and mode) in this video.