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Median of Discrete Frequency Distribution

To calculate the median we proceed as follows:

1. Calculate the less than cumulative frequencies.

2. Find ,where .

3. Note the cumulative frequency just more than .

4. The corresponding value of the variable is the Median.


Note:
Suppose appears as one of the cumulative frequency value then take the corresponding value of the observation and the value of the next observation. The average of these two numbers gives the median.

 

Example

 Calculate the median from the following distribution:

x

1

2

3

4

5

6

7

8

f

10

12

15

14

5

12

13

9

Solution

First we can find the cumulative frequency of the given distribution:

x

f

Cumulative frequency

1

10

10

2

12

10 + 12 = 22

3

15

22 + 15 = 37

4

14

37 + 14 = 51

5

5

51 + 5 = 56

6

12

56 + 12 = 68

7

13

68 + 13 = 81

8

9

81 + 9 = 90

 

90

 

  


 

x

f

Cumulative Frequency

1

10

10

2

12

10 + 12 = 22

3

15

22 + 15 = 37

4

14

37 + 14 = 51

5

5

51 + 5 = 56

6

12

56 + 12 = 68

7

13

68 + 13 = 81

8

9

81 + 9 = 90

 

 

We find the cumulative frequency just more than i.e, 45 is 51 and the value of

x corresponding to 51 is 4. Therefore, Median = 4.

 

 
Example

Calculate the median from the following frequency distribution:

Marks

19

28

27

32

37

41

42

24

Number of Students

7

12

22

14

13

14

19

19

 
Solution

Find the cumulative frequency

Marks

Number of Students

Cumulative frequency

19

7

7

24

19

7 + 19 =26

27

22

26 + 22 = 48

28

12

48 + 12 = 60

32

14

60 + 14 =74

37

13

74 +13 = 87

41

14

87 + 14 = 101

42

19

101 + 19 = 120

HereN is even

The median will be the average of the and  observations, so average of 30th and 31st observation i.e. .

Therefore the Median = 28.5.

 

Median of a Grouped or Continuous frequency distribution

To calculate the median of a grouped frequency distribution we proceed as follows

1. Calculate less than cumulative frequency.

2. Find .

3. Find the cumulative frequency just more than .

4. The corresponding class contains the median value and is called the median class.

5. The value of median is obtained by the formula .
Where

l = lower limit of the median class
n = number of observations
c.f = cumulative frequency of the class preceding the median class
f = frequency of the median class
h = class size or width of the median class


 

Example

Find the median for the following distribution:

Wages (in Rs)

0-10

10-20

20-30

30-40

40-50

Number of workers

5

7

10

8

5

 

Solution

Wages (in Rs)

Number of Workers

Cumulative Frequency

0-10

5

5

10-20

7

5 + 7 = 12

20-30

10

12 + 10 = 22

30-40

8

22 + 8 = 30

40-50

5

30 + 5 = 35

Here 

The cumulative frequency just more than 17.5 is 22

The median class is 20-30

l = 20, h = 10, c.f =12, f =10,

Using the formula,

 
  = 20 + 5.5
  = 25.2
 


 

Example

If the median of the distribution given below is 46, find the value of

 the missing frequencies.

Variable

10-20

20-30

30-40

40-50

50-60

60-70

70-80

Frequency

12

30

?

65

?

25

18

 
Solution

Variable

Frequency

Cumulative Frequency

10-20

12

12

20-30

30

12 + 30 = 42

30-40

42 + 

40-50

65

42 + + 65 = 107 +

50-60

107 + +

60-70

25

107 + + + 25 = 132  ++

70-80

18

132 + + + 18 = 150 + + 

 

229

 

Let  be the frequency of the class 30-40 and  be the frequency of 50-60

Then 

Since the given median 46 lies in the class 40 – 50, it is the median class.

Using Median formula,

Median = 

 





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