Title: The Median
Course: UGQuants (SSPS) > Section: Univariate Analysis

The Median

The median is the ‘middle’ value for a variable when all the values taken by all the cases are arranged in order. This is easiest to illustrate with a very simple example. Imagine a university tutorial class with only 7 students. Their grades for a course essay, arranged in ascending order, are as follows:

52, 55, 55, 63, 65, 68, 75

The median (‘middle’) essay grade is therefore 63. Where we have an even number of cases we will have two ‘middle’ values. If these are not the same, then we typically calculate the median by taking the mean of those two values.

While the median is easy to identify when we have only a small number of cases, this might not be so straightforward where we have a very large number of cases because we need to take into account the value for every single case, even when many cases take the same value. But provided we are able to arrange our data in ascending order for the variable in question, and our cases are numbered, finding the median should be fairly easy: we simply find the ‘middle’ case. In practice, calculating the median is even easier than this if we have our data in electronic form because computer software does it for us.

Since interval/ratio level variables have an inherent ranking we can use the median as a measure of central tendency for such variables. But why would we want to do this when we are able to use the mean with such variables? One key reason is that the mean is quite susceptible to influence from extreme values (often referred to as outliers). A good example of this is personal income. In any population a relatively small number of people earning very high incomes might contribute to a somewhat misleading representation of the ‘average’ income. We might say that these very high incomes ‘pull up’ the mean. The median is not susceptible to this kind of distortion because it is not based on the sum of all values for a variable.

As well as ratio level data like income, we can also calculate the median for ordinal level data. We may do this because ordinal data can also be rank ordered. So, although we typically have a much smaller range of values for ordinal variables than we would for ratio variables, the principle in calculating the median is exactly the same: we arrange all values for all cases in ascending (or descending) order, and the ‘middle’ value is the median.

For example:

These data provide immediate basic evidence about the level of people’s interest in politics in Scotland; how worried people in Britain are about immigrants; and how people in Britain feel about their own health.