Median

In this article, we will learn what the median is, how to find it within an even or odd set of numbers, and how it differs from the mean.

The median is a certain index in statistics that reveals a specific piece of data about the entire set.
It is no coincidence that it is called "median," it comes from the middle, as it represents the element that is exactly in the middle. Half of the numbers are above it and the other half below.
For example: What is the median in the following numerical set?
$2,3,4,5,6$

Solution: the median is $4$. The two numbers below: $2$ and $3$ and the two above it: $5$ and $6$.
The median shows us that $4$ is the center of the numerical set.

Finding the median in an odd set of numbers is quite easy
We just need to find a number that has the same amount of lower values as it does higher values.
We can act according to the formula (when the number of numbers is odd) and thus obtain the position that the median occupies.
Observe: the formula does not return the median, but the place it occupies.
$\frac{N+1}{2} = Position~of~the~median$
$N$ ->   Number of elements in the set
If there are few elements in the set, we will not need the formula and can act in the following way:

First step -> Arrange the numbers in ascending order.
Second step -> Locate the number that is in the middle, separating two groups that have the same number of elements.

What is the median in the following set:
$23,10,11,24,13$?

Solution:
First step ->
Let's arrange the numbers in ascending order.

We will obtain:
$10,11,13,23,24$
Second step -> Let's locate the middle number.

We will obtain:
The median is $13$. It is in the center - below it there are $2$ numbers and above it there are another $2$ numbers.

What would happen if we were asked to find the median within a very large set of numbers? For example, $135$ numbers?
In such a case, we should use the formula to find the position where the median is located.

What is the position of the median in a set that has $135$ numbers?

Solution:
Let's use the formula:
$\frac{135+1}{2}=68$
The position of the median is $68$.

When we are presented with a set with an even number of elements, we cannot locate the center one at first glance, since there is no element that has the same number of numbers above it as it does below.
So, what is done?
The median in a set with an even number of numbers is the average of the 2 elements that occupy the central positions.
That is, if there are $N$ numbers in the set and $N$ is an even number, the median will be the average of the two elements that occupy the positions:
$\frac{N}{2}$  and $\frac{N+2}{2}$

What is the median in the following set:
$4,7,8,9,10,13$?

Solution:
Observe: If the numbers are not arranged in ascending order, first, we will arrange them.
We will find the $N$ number of elements in the set -> $6$
The median will be the average of the number in the third position ${6\over2} = 3$ and the number in the fourth position $\frac{6+2}{2}=4$.
That is, the median is the average of the elements $8$ and $9$. We will obtain:
$\frac{8+9}{2}=8.5$
The median is $8.5$.

What would be the median in a set that has $400$ numbers?

Solution:
The median will be the average of the two numbers that occupy the positions:
${400 \over 2} = 200$
and
$\frac{400+2}{2}=201$
Note: We are not actually calculating the average of $200$ and $201$. These are just the positions that the elements whose values we will take to calculate the average and then, obtain the median.

The median is not affected by extreme numbers while the mean is influenced by all the numbers in the set.
Is it an advantage or disadvantage? We cannot know. It depends on the given set of numbers. Therefore, we cannot argue that one index is better than the other absolutely.