Chapter 25.9 - Median of a Data

In the previous section we completed the discussion on Mean. In this section we will see Median.

Median

We are going to learn about median and it's significance. But first we will learn how to calculate the median:
■ Any list of numbers will have a median. It is the 'middle number'.
■ If the list contains an odd number of numbers, one of the number already in the list will be the median. 
• This number will be situated at the exact middle, 
      ♦ when the whole list is sorted in ascending or descending order.
■ If the list contains an even number of numbers, the median will not be already present in the list. We will have to do some calculations to find it.
• This is because, there will be two numbers at the exact middle,
    ♦ when the whole list is sorted in ascending or descending order
• The median is the mean of those two numbers at the middle

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Let us elaborate on the above two cases with the help of examples: fig.25.22

Case 1: When the list has an odd number of numbers

1. Consider fig. 25.22(a) below. It has 13 (an odd number) values sorted in ascending order. 

Fig.25.22

2. Each can be denoted as x_i. So we have x₁, x₂, x₃, . . . , upto x_n. where n = 13

3. 'i' denotes the position of each value in the sorted list

4. We want the value of 'i' of the median. For calculating that value of 'i', we adopt the following procedure:

(a) There are equal number of values on either side of the median. Let this equal numbers be 'x'

(b) So we get: 

x + 1 ( the median) + x = n 

⇒ 2x + 1 = n. ⇒ 2x = n-1 ⇒ x = ^(n-1)⁄₂

(c) So there are ^(n-1)⁄₂ numbers on the left side of the median

(d) That means, the 'i' value of the number just to the left of the median = ^(n-1)⁄₂

(e) So the 'i' value of the median = ^(n-1)⁄₂  + 1 = ^(n-1+2)⁄₂  = ⁽ⁿ⁺¹⁾⁄₂.

5. When n = 13, we get 'i' value of the median = ⁽ⁿ⁺¹⁾⁄₂  = ⁽¹³⁺¹⁾⁄₂ = ¹⁴⁄₂ = 7

6. From the fig.25.22(a), we can see that the 7^th value is indeed the median

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So when the number of values (n) in the list is an odd number, we can find the median by the following procedure:

1. Sort the list in ascending or descending order

2. Take out the value whose i = ⁽ⁿ⁺¹⁾⁄₂

3. This value is the median

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Case 2: When the list has an even number of numbers

1. Consider fig. 25.22(b) above. It has 12 (an even number) values sorted in ascending order. 

2. Each can be denoted as x_i. So we have x₁, x₂, x₃, . . . , upto x_n. where n = 12

3. 'i' denotes the position of each value in the sorted list

4. There are two values at the 'exact middle'. We will call them the 'middle pair'. For any sorted list having an even number of values, there will be a 'middle pair'.  

5. We want the value of 'i' for both the members of that pair. For calculating those value of 'i', we adopt the following procedure:

(a) There are equal number of values on either side of the middle pair. Let this equal numbers be 'x'

(b) So we get: 

x + 2 ( the middle pair) + x = n 

⇒ 2x + 2 = n. ⇒ 2x = n-2 ⇒ x = ^(n-2)⁄₂

(c) So there are ^(n-2)⁄₂ numbers on the left side of the middle pair

(d) That means, the 'i' value of the number just to the left of the pair = ^(n-2)⁄₂

(e) So the 'i' value of the first member of the pair = ^(n-2)⁄₂  + 1 = ^(n-2+2)⁄₂  = ⁿ⁄₂.

(f) So the 'i' value of the second member of the pair = ⁿ⁄₂  + 1 .

6. When n = 12, we get:

• 'i' of the first member of the middle pair = ⁿ⁄₂ = ¹²⁄₂ = 6

• 'i' of the second member of the middle pair = (ⁿ⁄₂  + 1) = 7

7. So the 6^th and 7^th values form the middle pair. From fig.25.22(b), we can see that, this is indeed true. 

8. Once we get the i values of both the members of the middle pair, we can take them out from the list

9. Then we calculate the mean of the two members. This mean is the median of the whole list

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So when the number of values (n) in the list is an even number, we can find the median by the following procedure:

1. Sort the list in ascending or descending order

2. Take out the value whose i = ⁿ⁄₂

_{3. Take out the value whose i = ⁽ⁿ⁺²⁾⁄2}

4. Calculate the mean of the values in (2) and (3). This mean is the median of the whole list

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Now we will see some solved examples on the calculation of median

Solved example 25.22

The heights (in cm) of 9 students of a class are as follows:

Find the median of this data.

Solution:
There are 9 values. So n = 9. It is an odd number. We can use the procedure that we wrote earlier:

1. Sort the list in ascending or descending order

2. Take out the value whose i = ⁽ⁿ⁺¹⁾⁄₂

3. This value is the median
• The list sorted in ascending order is:

The numbers in yellow colour shows the sequence 'i' • i = ⁽ⁿ⁺¹⁾⁄₂ = ⁽⁹⁺¹⁾⁄₂ = ¹⁰⁄₂ = 5
• So the 5^th value 149 is the median

Solved example 25.23
The points scored by a Kabaddi team in a series of matches are as follows:  

Find the median of the points scored by the team.
Solution:
There are 16 values. So n = 16. It is an even number. We can use the procedure that we wrote earlier:

1. Sort the list in ascending or descending order

2. Take out the value whose i = ⁿ⁄₂

_{3. Take out the value whose i =} (ⁿ⁄₂  + 1)

4. Calculate the mean of the values in (2) and (3). This mean is the median of the whole list
• The list sorted in ascending order is:

The numbers in yellow colour shows the sequence 'i'
• i = ⁿ⁄₂ = ¹⁶⁄₂ = 8. So the first member of the middle pair = 10
• (ⁿ⁄₂  + 1) = 9. So the second member of the middle pair = 14
• Median of the list = Mean of 10 and 14 =  ^(10+14)⁄₂ = ²⁴⁄₂ = 12

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We have completed the discussion on Median. In the next section we will see Mode

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