Median

Median is the most middle value in the arrayed data. It means that when the data are arranged, median is the middle value if the number of values is odd and the mean of the two middle values, if the numbers of values is even. A value which divides the arrayed set of data in two equal parts is called median, the values greater than the median is equal to the values smaller than the median. It is also known as a positional average. It is denoted by \widetilde X read as X- tilde.

Median from Ungrouped Data:
Median = Value of {\left( {\frac{{n + 1}}{2}} \right)^{th}}item

Example:
Find the median of the values 4, 1, 8, 13, 11

Solution:
Arrange the data 1, 4, 8, 11, 13

Median = Value of {\left( {\frac{{n + 1}}{2}} \right)^{th}}item

Median = Value of {\left( {\frac{{n + 1}}{2}} \right)^{th}}item = {\left( {\frac{6}{2}} \right)^{th}} = {3^{th}}item

Median = 8

Example:
Find the median of the values 5, 7, 10, 20, 16, 12

Solution:
Arrange the data 5, 7, 10, 12, 16, 20

Median = Value of {\left( {\frac{{n + 1}}{2}} \right)^{th}}item

Median = Value of {\left( {\frac{{n + 1}}{2}} \right)^{th}}item = {\left( {\frac{7}{2}} \right)^{th}} = {3.5^{th}}item

Median = \frac{{10 + 12}}{2} = 11

Median from Ungrouped Data:

The median for grouped data, we find the cumulative frequencies and then calculated the median number\frac{n}{2}. The median lies in the group (class) which corresponds to the cumulative frequency in which \frac{n}{2} lies. We use following formula to find the median.

Median = l + \frac{h}{f}\left( {\frac{n}{2} - c} \right)

Where
l= Lower class boundary of the model class
f= Frequency of the median class
n = \sum f = Number of values or total frequency
c= Cumulative frequency of the class preceding the median class
h= Class interval size of the model class

Example:
Calculate median from the following data.

Group
60 – 64
65 – 69
70 – 74
75 – 79
80 – 84
85 – 89
Frequency
1
5
9
12
7
2

Solution:

Group
f
Class Boundary
Cumulative Frequency
60 – 64
1
59.5 – 64.5
1
65 – 69
5
64.5 – 69.5
6
70 – 74
9
69.5 – 74.5
15
75 – 79
12
74.5 – 79.5
27
80 – 84
7
79.5 – 84.5
34
85 – 89
2
84.5 – 89.5
36

Median = l + \frac{h}{f}\left( {\frac{n}{2} - c} \right)  \because {\left( {\frac{n}{2}} \right)^{th}}item  = {\left( {\frac{{36}}{2}} \right)^{th}} = {18^{th}}item

Median = 74.5 + \frac{5}{{12}}\left( {18 - 15} \right) = 74.5 + \frac{5}{{12}}\left( 3 \right) = 75.75

Median from Discrete Data:
When the data follows the discrete set of values grouped by size. We use the formula {\left( {\frac{{n + 1}}{2}} \right)^{th}}item for finding median. First we form a cumulative frequency distribution and median is that value which corresponds to the cumulative frequency in which {\left( {\frac{{n + 1}}{2}} \right)^{th}}item lies.

Example:
The following frequency distribution is classified according to the number of leaves on different branches. Calculate the median number of leaves per branch.

No of Leaves
1
2
3
4
5
6
7
No of Branches
2
11
15
20
25
18
10

Solution:

No of Leaves
X
No of Branches
f
Cumulative Frequency C.F
1
2
2
2
11
13
3
15
28
4
20
48
5
25
73
6
18
91
7
10
101
Total
101

Median = Size of {\left( {\frac{{n + 1}}{2}} \right)^{th}}item  = \frac{{101 + 1}}{2} = \frac{{102}}{2} = 51item

Median = 5 because {51^{th}}item lies corresponding to 5