Median – Definition

Median is the middle value of a given distribution or group of numbers in their ascending order.

OR 

in simplest way, we can say that the median is the midpoint of the given set of data.

Median = Middle number 

Example: If we have numbers like 1, 2, 3, 4, 5.

The number in the middle of the series is the median.

Here we see that, the data are in ascending order and middle number of series is 3, so the median of the series is 3.

The median of 1, 2, 3, 4, 5 is 3.

Let’s understand the concept of median properly in brief with more examples.

Example: Numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9.

Here we see that numbers are in ascending order and middle number of series is 5, so the median of the series is 5.

Example: Numbers are 11, 24, 35, 43, 57, 61, 75, 82, 90.

Here we see that data are in ascending order and middle number of series is 57, so the median of the series is 57.

The median of 11, 24, 35, 43, 57, 61, 75, 82, 90 is 57.

Example- 10, 16, 22, 31, 47, 58, 66, 72, 84, 87, 98.

Here we see that data are in ascending order and middle number of series is 58, so the median of the series is 58.

How to find the median value

Step 1: Count the total number of the given observation.

Step 2: Arrange the numbers in ascending order.

Step 3: If the number of observations is odd , the middle number of observation is median and find by formula.

Median = (n + 1)/2 -th term , where n is odd.

Example- 9, 4, 2, 6, 5, 1, 12.

Step 1: Count the total number of the given observation.
Here ,number of observations are 7, that means the set of data has odd number of observations.

Here we notice that numbers are not arranged in ascending order. 

Step 2: Arrange the numbers in ascending order. So numbers in ascending order are 1, 2, 4, 5, 6, 9, 12.

Step 3: If the number of observations is odd , the middle number of observation is median and determined by using the formula.

Step 4: If the number of observations is even, the middle number of observation is median and is determined using the formula.

When n is odd
Median = (n + 1)/2 -th term

Median = (7 +1)/2 -th term, where n is 7, that is odd.

Median = 8/2 -th term

median = 4 -th term

median = 5

hence median is 5.

                        When n is even

Median = {(n/2)th + (n/2+1)th} term/2 

Example- 2, 4, 6, 9

Number of observations are 4, means the set of numbers has even number of observations.

Numbers are arranged in ascending order, but in this case, there are two middle terms. Hence we need to find the middle pair of numbers and add them together and divide result by 2 ( or simple average of the two middle terms).

If the number of observations are even, then the median is the average of the two middle terms.

Therefore, the median is average of two middle terms

∴ median = (4 + 6)/2

median = 10/2

median = 5

Note:  5 is not in the observations, but it’s ok because half of the numbers in the observations are less than 5 and half of numbers are greater than 5.

Example: 12, 29, 34, 46, 49, 51, 64, 72, 86, 98.

Number of observations are 10, means the set of data has even number of observations.

Data are arranged in ascending order, but there are two middle terms as 49 and 51.

If the number of observations are even then the median is the average of the two middle terms.

Therefore, the median is average of two middle terms are (49 + 51)/2

median = 100/2

median = 50

Example: Numbers are 20, 25, 32, 39, 40, 46, 57, 67, 79, 87, 89, 90, 93, 96.

Here number of observations are 14, means the set of data has even number of observations.

Data are arranged in ascending order, but there are two middle terms, 57 and 67.

Therefore, the median is average of two middle terms are (57 + 67)/2

median = 124/2

median = 62

Now we shall try to determine median using the formula:

Median = {(n/2)th + (n/2+1)th}term/2 

Example: Numbers are 11, 24, 27, 32, 38, 40, 45, 58, 68, 79, 87, 89, 90, 93, 96, 98.

There are number of observations are 16, means the set of data has even number of observations.

Here n = 16 that is even, put the value in the formula,

Median = {(16/2)th + (16/2 +1)th}/2 

 median = {8th + 9th}/2 

median = {58 + 68}/2 

median = {126}/2 

median = 63 

Example: Numbers are 11, 24, 26  28, 30, 32, 38, 40, 46, 58, 68, 79, 87, 89, 90, 93, 96, 98.

There are number of observations are 18, means the set of data has even number of observations.

Here n = 18 that is even, put the value in the formula

Median = {(18/2)th + (18/2 +1)th}term/2 

Median = {9th + 10th}term/2 

median = {46 + 58}/2 

median = {104}/2 

median = 52