Median

Descriptive statistics » Measures of central tendency » Median

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The median is the mid-range value of a distribution of scores for a given variable. The median, thus, divides such distribution of scores exactly in half. The median can be considered as a non-parametric measure of central tendency and as the proper average when doing statistical analysis at ordinal levels.

Interpreting a median

There seems to be no standard notation for the median, thus, if one if used, it would be defined as such in the published material. 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 could be 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, imaging that you were told that a group of schoolchildren were given a test of happiness, ranging from 1, 'Very unhappy' to 5, 'Very happy', and that the median for the group was 4, 'Happy'.

This information literally tells us 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'. Assuming that each child contributed one score only, we can thus 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 mid-range 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 non-normally distributed interval or 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.
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.