20120816 - Median

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# The median

In statistics, the median is the value that marks the middle of a ranked distribution of scores. It also divides such distribution exactly in half, so that 50% of the distribution is below the median and 50% of it is above the median. Thus, the median is a measure of central tendency, and it is also the proper average when doing statistical analyses at ordinal levels.

## Interpreting a median

There seems to be no standard notation for the median, thus it may be annotated as 'median', 'm', 'Md', etc. In any case, the median is interpreted as the "middle value" in a dataset, or as the "average value" in the distribution of an ordinal variable. A more formal definition is the following: the median is that value at the 50th 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)1.

Following from above definitions, it is clear that the median represents the middle score in a ranked distribution and it is not affected by the actual value of particular scores (ie, scores are, simply, ordered in a ranked sequence). This property makes the median both an average not influenced by extreme cases and a good indicator to know something about, 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 about two things: firstly, that 4, 'Happy' was the ordinal average of the distribution of scores given by those children, and secondly, that at least 50% of the scores were either 4, 'Happy' or 5, 'Very happy' or both. Of course, another way of looking at it is that 50% of the remaining scores may have ranged from 1, 'Very unhappy', to either 3, 'Neither happy nor unhappy', or, even, 4, 'Happy'. Whatever the point-of-view chosen, however, the median is closer to the upper end of the happiness scale. Thus, assuming that each child contributed one score only, you could correctly infer that these schoolchildren were overall happy, with at least half the class reporting being happy or very happy.

When used in a quasi-inferential manner, the median informs of the middle value in a future dataset.

For example, from knowing above results, and assuming that you are unable to gather new data regarding the happiness of schoolchildren, you could "second-guess" that the same children would be mostly happy next year, that children of similar age would be mostly happy in the same school, or that children of similar age would be mostly happy in other schools.

## Properties of the median

• The median is a descriptive statistic that informs about the middle score in a ranked distribution1.
• When samples are normally distributed, the median equals both the mode and the mean. Otherwise, it may differ from those two measures, sometimes greatly.
• Thus, if you are given both the median and the mean, you can assess whether the distribution of scores is skewed or not (thus, whether it is normal on that account). A great difference between both measures in either direction (ie, a mean which is either sensibly smaller or bigger than the median) indicates that the distribution is skewed and, thus, non-normal. In such cases, the median may be a better indicator of central tendency.
• Good measures of dispersion around the median are the interquartile range and the standard percentile range.
• A good graphic for the median is the boxplot.
Footnotes
1. 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 half. This is not the case if scores are not independent, of course.

# Want to know more?

RobertNiles.com - Median
Stats4Students.com
This page contains an article explaining more in detail measures of central tendency. The article is, LUND RESEARCH (2007). Measures of central tendency. Stats4Students.com.
Wiki of Science - Descriptive statistics
This Wiki of Science page provides you access to more descriptive statistics.
Wiki of Science - Median (calculation)
This Wiki of Science page provides you with the tools to identify the median.

## Editor

Jose D PEREZGONZALEZ (2012). Massey University, Turitea Campus, Private Bag 11-222, Palmerston North 4442, New Zealand. ().

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page revision: 1, last edited: 16 Aug 2012 23:45