Median

Median describes position of a specific term.

Median is defined as the middle term of numbers arranged in order, either ascending or descending.

Example:

What is the median of the following numbers?

7, 5, 2, 3, 1, 4, 6?

Answer:

First, write the given numbers in order.

It is 1, 2, 3, 4, 5, 6 and 7.

Locate the middle term. It is 4.

Hence, median is 4.

The given terms

1, 2, 3, 4, 5, 6 and 7 are in Arithmetic sequence.

Now, what is the mean of numbers in arithmetic sequence?

It is (first term + last term)/2

Therefore,

Mean is (1 + 7)/2 = 8/2 = 4

Make the following Note:

Note:

In arithmetic sequence, mean and median are equal.

The numbers 1, 2, 3, 4, 5, 6 and 7 are in arithmetic sequence and therefore, both the mean and the median are equal i.e. 4.

Example:

What is the median of n consecutive numbers whose average is 2n² + 4n + 1?

Answer:

Consecutive numbers are in arithmetic sequence, as the difference between any two consecutive numbers is same i.e. 1.

Therefore, mean and median will be equal.

As mean is 2n² + 4n + 1, so median will also be 2n² + 4n + 1.

How to find the Middle Term:

By definition, median is middle term.

Now, middle term will depend on number of terms N, which can be either even or odd.

Case 1:

When number of terms N is odd:

If N is odd, only one middle term exists, and it is
(N + 1)/2 th term.

Suppose there are 51 terms.

Then the middle term is (51 + 1)/2 th term, i.e. 26^th term.

Whichever term is in the 26^th place or position is the middle term and will therefore be the Median.

Case 2:

When number of terms N is even:

There will be two middle terms when the number of terms N is an even number.
Suppose there are 50 terms.

Since 50 is an even number, the two middle terms will be

N/2th term and (N/2 + 1)th term

So, in 50 terms,

the two middle terms are 25^th and 26^th terms.

Median is next found by calculating the average of the two middle terms, i.e. whichever numbers are in the 25^th and 26^th places.