How do you find the median of a set of values when there is an even number of values?

Question

Nora Amin · Accepted Answer

The average of the two "central" values, or the value that divides the set so that exactly half of the numbers are less than it and exactly half are greater than it, is the definition of the median of a set of numbers with an even number of numbers. If #S = { 2, 4, 5, 7, 9, 14, 17, 25}# (note that for simplicity I have re-arranged the elements of this set to be in ascending order). The median of #S# would be 
#(7 + 9)/2 = 8#
Half of the values of #S#, namely #{2, 4, 5, 7}#, are less than the median #(8)# 
and 
half of the values of #S#, namely #{9, 14, 17, 25}#, are greater than the median #(8)#  Remember that a set of numbers can have duplicate values; for instance, if the numbers are test scores from a class. One issue could occur if the median value appears more than twice within the set. For example:
#T = { 2, 5, 5, 6, 6, 6, 6, 9}# In the sense that no value divides the set into two equal-sized subsets, there is no true median value. In this case the median would normally be taken to be #6# but care would need to be taken in its application. Luckily, this typically only occurs with reasonably large sets where there is no discernible difference in size between the "less than" and "greater than" subsets.