Median

Median of any distribution is that value that divides the distribution into two equal parts such that the number of observations above it is equal to the number of observations below it. Thus, the median is called the central value of any given data either grouped or ungrouped.

Median of Ungrouped Data

To calculate the Median, the observations must be arranged in ascending or descending order. If the total number of observations is N then there are two cases

Case 1: N is Odd

Median = Value of observation at [(n + 1) ÷ 2]^th Position

When N is odd the median is calculated as shown in the image below.

Case 2: N is Even

Median = Arithmetic mean of Values of observations at (n ÷ 2)^th and [(n ÷ 2) + 1]^th Position

When N is even the median is calculated as shown in the image below.

Example 1: If the observations are 25, 36, 31, 23, 22, 26, 38, 28, 20, 32 then the Median is given by 

Arranging the data in ascending order: 20, 22, 23, 25, 26, 28, 31, 32, 36, 38

N = 10 which is even then

Median = Arithmetic mean of values at (10 ÷ 2)th and [(10 ÷ 2) + 1]th position

⇒ Median = (Value at 5th position + Value at 6th position) ÷ 2

⇒ Median = (26 + 28) ÷ 2

⇒ Median = 27

Example 2: If the observations are 25, 36, 31, 23, 22, 26, 38, 28, 20 then the Median is given by

Arranging the data in ascending order: 20, 22, 23, 25, 26, 28, 31, 36, 38

N = 9 which is odd then

Median = Value at [(9 + 1) ÷ 2]^th position

⇒ Median = Value at 5^th position

⇒ Median = 26

Median of Grouped Data

Median of Grouped Data is given as follows:

[Tex]\bold{Median =l+ \frac{N/2 – c_f}{f} \times h}[/Tex]

Where,

• l is the lower limit of median class,
• n is the total number of observations,
• c_f is the cumulative frequency of the preceding class,
• f is the frequency of each class, and 
• h is the class size.

Example: Calculate the median for the following data.

Solution:

Create the following table for the given data.

Class	Frequency	Cumulative Frequency
10 – 20	5	5
20 – 30	10	15
30 – 40	12	27
40 – 50	8	35
50 – 60	5	40

As n = 40 and n/2 = 20,

Thus, 30 – 40 is the median class.

l = 30, c_f = 15, f = 12, and h = 10

Putting the values in the formula  [Tex]\bold{Median =l+ \frac{N/2 – c_f}{f} \times h} [/Tex]

Median = 30 + (20 – 15)/12) × 10

⇒ Median = 30 + (5/12) × 10

⇒ Median = 30 + 4.17

⇒ Median = 34.17

So, the median value for this data set is 34.17

Central Tendencies in Statistics are the numerical values that are used to represent mid-value or central value a large collection of numerical data. These obtained numerical values are called central or average values in Statistics. A central or average value of any statistical data or series is the value of that variable that is representative of the entire data or its associated frequency distribution. Such a value is of great significance because it depicts the nature or characteristics of the entire data, which is otherwise very difficult to observe.