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. 26th term.

Whichever term is in the 26th 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 25th and 26th terms.

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