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.

## Definition of Median in Statistics

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.

In the data set displayed above, the median is 7. It can be seen on either side of mean there are 6 values.

## Median Statistics Formula

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.

## Examples of Median

### 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.

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

Median age of the inmates = 72.

Question 2: Find the Median of following frequency distribution

 Prize of a dressitem 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 ofitems 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.

### Mode

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