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.
Â
Â 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 
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 
Â 
90 
Â 
Â Â
Â
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 
Â
Â
We find the cumulative frequency just more thanÂ i.e, 45 is 51 and the value of
xÂ corresponding to 51 is 4. Therefore, Median = 4.
Â
ÂCalculate 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 Â

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 
Here,Â NÂ is even
The median will be the average of theÂ andÂ Â observations, so average of 30^{th}Â and 31^{st}Â observation i.e.Â .
Therefore the Median = 28.5.
Â
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 
Â
Find the median for the following distribution:
Wages (in Rs) 
010 
1020 
2030 
3040 
4050 
Number of workers 
5 
7 
10 
8 
5 
Â
Wages (in Rs) 
Number of Workers 
Cumulative Frequency 
010 
5 
5 
1020 
7 
5 + 7 = 12 
2030 
10 
12 + 10 = 22 
3040 
8 
22 + 8 = 30 
4050 
5 
30 + 5 = 35 
HereÂ
The cumulative frequency just more than 17.5 is 22
The median class is 2030
l =Â 20, h =Â 10, c.f =12, f =10,
Using the formula,
Â  
Â  = 20 + 5.5 
Â  = 25.2 Â 
Â
If the median of the distribution given below is 46, find the value of
Â the missing frequencies.
Variable 
1020 
2030 
3040 
4050 
5060 
6070 
7080 
Frequency 
12 
30 
? 
65 
? 
25 
18 Â

Variable 
Frequency 
Cumulative Frequency 
1020 
12 
12 
2030 
30 
12 + 30 = 42 
3040 
42 +Â 

4050 
65 
42 +Â + 65 = 107 + 
5060 
107 +Â + 

6070 
25 
107 +Â +Â + 25 = 132Â Â ++ 
7080 
18 
132 +Â +Â + 18 = 150 +Â +Â 
Â 
229 
Â 
LetÂ Â be the frequency of the class 3040 andÂ Â be the frequency of 5060
ThenÂ
Since the given median 46 lies in the class 40 â€“ 50, it is the median class.
Using Median formula,
Median =Â
Â