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

Feb 12, 2015

The median of a set of numbers containing an even number of numbers is defined to be the average of the two "central" values;
it is a value which splits the set so exactly half the numbers in the set are less than it and exactly half are greater than it.

If $S = \left\{2 , 4 , 5 , 7 , 9 , 14 , 17 , 25\right\}$ (note that for simplicity I have re-arranged the elements of this set to be in ascending order).

The median of $S$ would be
$\frac{7 + 9}{2} = 8$
Half of the values of $S$, namely $\left\{2 , 4 , 5 , 7\right\}$, are less than the median $\left(8\right)$
and
half of the values of $S$, namely $\left\{9 , 14 , 17 , 25\right\}$, are greater than the median $\left(8\right)$

One problem can arise if the median value occurs more than twice within the set. (Remember that a set of numbers can have duplicate values; for example, if the numbers are test scores from a class).

For example:
$T = \left\{2 , 5 , 5 , 6 , 6 , 6 , 6 , 9\right\}$

There is no true median value (in the sense that there is no value that separates the set into two subsets of equal size).

In this case the median would normally be taken to be $6$ but care would need to be taken in its application.

(Fortunately this normally only happens with fairly large sets where the difference in size between the "less than" subset and the "greater than" subset is not significant).