Median

This article is about the statistical method, for alternative meanings see Median.

In statistics, the median is that value that separates the highest half of the sample from the lowest half. More precisely 1/2 of the population will have values less than or equal to the median and 1/2 of the population will have values equal to or greater than the median. To find the median, arrange all the observations from lowest value to highest value and pick the middle one. If there are an even number of observations, take the mean of the two middle values. When we use the median to describe what the observations have in common, there are several choices for a measure of variability, the range, the interquartile range, and the absolute deviation. Since the median is the same as the second quartile, its calculation is illustrated in the article on quartiles.

The median is primarily used for skewed distributions, which it represents more accurately than the arithmetic mean. Consider the set {1, 2, 2, 2, 3, 9}. The median is 2 in this case, as is the mode, and it might be seen as a better indication of central tendency than the arithmetic mean of 3.166....

The median is also the central point which minimises the average of the absolute deviations; in the example above this would be (1 + 0 + 0 + 0 + 1 + 7)/6 = 1.5 using the median, while it would be 1.944... using the mean.

Even though sorting n items takes in general O(n log n) operations, by using a recursive "Divide-and-Conquer" algorithm the median of n items can be computed with only O(n) operations.

Calculation of medians is a popular technique in summary statistics and summarizing statistical data, since it is simple to understand and easy to calculate, while also giving a measure that is more robust in the presence of outlier values than is the mean. The difference between the median and the mean is less than or equal to one standard deviation.

Median

See also

See also