Lesson Basics of Median

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Median is one of the statistics used for estimating the central value of a series. The median can be defined as that value which separates the upper half of a series from the lower half when arranged in ascending or descending order. The median is thus the middle value of the series when arranged in increasing or decreasing order.

In case there are an even number of elements in the series, there would be two central value: n/2 and (n+2)/2. In this case the median can be defined in two ways:
1. Median would be the mean of the two central values.
2. There would be two median, an upper median and a lower median.

Median is especially useful compared to median in situations where there may be a few elements in the series which are offset from the remaining values by a large amount. Such large deviations can effect the mean of the series, but the median would still be a good estimate of the central value of the series. For example, consider a set of values {2,3,3,2,1,3,21}. The mean of the series would be 5, while the median would be 3, which is closer to most values of the set.

The method for calculating the median is as follows:
1. Sort the series in ascending order.
2. If there are an odd number of elements in the series, the element at the (n+1)/2 position in the sorted series would be the median of the series.
3. In case there are an even number of elements in the series, the element at n/2 position forms the lower mean and the element at (n+2)/2 position in the sorted series forms the upper median. A single median would be defined as the mean or average of the two medians.

Example 1: Find the median of the following series: {1, 12, 17, 5, 9}.
Solution: The series sorted in ascending order would become {1, 5, 9, 12, 17}. The number of elements in the series is 5, which is odd and hence the median would be the element on (5+1)/2 = 3rd position which is 9.

Example 2: Find the median of the following series: {1, 11, 7, 15, 9, 12}.
Solution: The series sorted in ascending order would become {1, 7, 9, 11, 12, 15}. The number of elements in the series is 6, which is even and hence there would be two medians. The lower mean would be the element on (6)/2 = 3rd position which is 9 and the upper median would be the element on the (6+2)/2 = 4th position which is 11. The single median would be the mean of the two medians which would be (9+11)/2 = 10.
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