Median is a measure which is easy to understand and easy to determine. It is the halfway point in a data set and hence known as the positional average. Since it is not dependent on the actual values in the data set, it is less affected by extreme values. Against these merits, median is not based on all values and is not capable of algebraic treatments. It is also affected by sampling fluctuations.
Median is also a measure of position along with quartiles and Deciles. Median can also be understood as the second quartile or 50th percentile.

If the data values of a numerical data set is arranged in ascending or descending order of magnitude, the middle most item is called the median of the data set.

In numerical data distributions one half of the items are less than the median value and the other half is greater than the median value. The symbol used for median is MD.

Median Statistics
In the data set displayed above, the median is 7. It can be seen on either side of mean there are 6 values.
The following two observations lead to the formula for median:
  1. When the number of values in the data set is odd, the median is the middle value when the data is ordered. Thus the median is the $(\frac{n+1}{2})^{th}$ observation.
  2. When the number values in the data set is even, there are two middle values in the ordered array and median is the average of these two middle values. Thus the median is the average of the $(\frac{n}{2})^{th}$ and $(\frac{n}{2}+1)^{th}$ observations.

When the data is given in frequency distribution the median class is found by using the above formula. The actual value of the median is then determined proportionately.


Solved Examples

Question 1: The ages of 15 old people living in an old age Home are 65, 62, 78, 82, 89, 90, 73, 69, 70, 70, 71, 72, 78, 68 and 72.  Find the median of the data set.
Solution:
 
Step 1. Arrange the data in ascending order.

     Median Example

     Step 2. The number of data values n =15. Hence Median is the 8 value in the ordered array.

     Median Examples

                  Median age of the inmates = 72.
 

Question 2: Find the Median of following frequency distribution
    
Prize of a dress
item in dollars
60
 75   90   100 
 110 
  120 
Number of
items sold
 20
 18  22  18   16  18

Solution:
 
We rewrite the table including a column for cumulative frequency.
  
Prize
  x
Number of
items sold
       f
 Cumulative
 frequency
 
  ← Median class containing the
       56th and 57th items
  60       20
      20
  75       18
      38
  90       22
      60
 100       18
      78
 110      16
      94
 120      18
     112
  N = 112 

The total number of items sold N = 112.  Hence the Median is the average of $\frac{112}{2}$ th item and $\frac{112}{2}$ + 1 th item in the order. That is the average of the 56th and 57th items.  The row containing the cumulative frequency 60 will contain these two items whose prizes are both 90 dollars.
Hence the median prize of the items sold = $\frac{90+90}{2}$ = 90 dollars.