Median

1. MEDIAN
NADEEM UDDIN
ASSOCIATE PROFESSOR
OF STATISTICS

2. MEDIAN:
The median of a set of
observations is the value that
falls in the middle when the
observation are arranged in order
of magnitude
or the median is a value at or
below which 50% of the ordered
data lie.

3. Calculating the Median:
In case of ungroup data, the median can be
found by the following formula.
𝑀𝑒𝑑𝑖𝑎𝑛 =
𝑛 + 1
2
𝑡ℎ 𝑣𝑎𝑙𝑢𝑒, 𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑
𝑀𝑒𝑑𝑖𝑎𝑛 =
𝑛
2
𝑡ℎ 𝑣𝑎𝑙𝑢𝑒 +
𝑛 + 2
2
𝑡ℎ 𝑣𝑎𝑙𝑢𝑒
2
, 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛
n = Number of Terms or Observation

4. Example-33:
The wages of 7 workers in rupees are 3000,
4500, 5000, 2500, 4000, 2000, 3200.
Solution:
1st, we arranging the observation in ascending
order. We get 2000, 2500, 3000, 3200, 4000,
4500, 5000
Median = The value of th
2
1n





 
term
Median = The value of th
2
17





 
term
= The value of 4th
term
Median = Rs. 3200

5. Example-34:
The minimum temperature in Quetta for the first 10
days of February was 10, 10, 0, 4, 3, 8, 4, 3, 5, 1
Solution:
1st, we arranging the observation in ascending order.
We get 10, 8, 4, 3, 0, 1, 3, 4, 5, 10.
𝑀𝑒𝑑𝑖𝑎𝑛 =
𝑛
2
𝑡ℎ 𝑣𝑎𝑙𝑢𝑒 +
𝑛 + 2
2
𝑡ℎ 𝑣𝑎𝑙𝑢𝑒
2
𝑀𝑒𝑑𝑖𝑎𝑛 =
10
2
𝑡ℎ 𝑣𝑎𝑙𝑢𝑒+
10+2
2
𝑡ℎ 𝑣𝑎𝑙𝑢𝑒
2
𝑀𝑒𝑑𝑖𝑎𝑛 =
5𝑡ℎ 𝑣𝑎𝑙𝑢𝑒+6𝑡ℎ 𝑣𝑎𝑙𝑢𝑒
2
𝑀𝑒𝑑𝑖𝑎𝑛 =
0+1
2
= 0.5

6. In case of group data with classes
consisting of single values, the
median can be found by the
following formula.
Median = The value of
𝑛
2
term
Where,
n = Sum of the frequencies

7. No. of
Chairs
25 30 40 43 50 60
No. of
Rooms
2 4 10 8 6 1
Example-35:
Given below is the frequency distribution of
number of chairs in different rooms of a
college. Find the median.

8. Solution:
Let x denote the number of chairs.
Median = The value of
31
2
term
Median = The value of 15.5th term
Median = 40
x F C.f
25
30
40
43
50
60
2
4
10
8
6
1
2
6
16
24
30
31
Sum=n =31

9. In case of group data with classes consisting of
range, the median can be found by the following
formula
𝑀𝑒𝑑𝑖𝑎𝑛 = 𝑙 +
ℎ
𝑓
𝑓
2
− 𝑐. 𝑓
Where,
l = Lower class boundary of the median
class
Median Class: The class corresponding to
the cumulative frequency in which
𝑓
2
lies.
h = Size of class interval of the median
class
f = Frequency of the median class
𝑓 = Sum of the frequencies
C.f = Cumulative frequency of the class

10. Example-37:
Find the median for the following data.
Classes 10 – 19 20 – 29 30 – 39 40 – 49 50 – 59
Frequenc
y
5 25 40 20 10
Solution:
Classes f C.B C.f
10 – 19
20 – 29
30 – 39
40 – 49
50 – 59
5
25
40
20
10
9.5 – 19.5
19.5 – 29.5
29.5 – 39.5
39.5 – 49.5
49.5 – 59.5
5
30
70
90
100
Sum 100

11. 1st, find the median class:
𝑓
2
𝑡ℎ term =
100
2
term = 50th term
50th term lies in the class 30 – 39.
Therefore, 29.5 – 39.5 is the
medianclass.
l = 29.5,
h = 39.5 – 29.5 = 10,
f = 40,
𝑓
2
= 50,
C.f =30

12. Substitute the values we get:
𝑀𝑒𝑑𝑖𝑎𝑛 = 𝑙 +
ℎ
𝑓
𝑓
2
− 𝑐. 𝑓
𝑀𝑒𝑑𝑖𝑎𝑛 = 29.5 +
10
40
50 − 30
𝑀𝑒𝑑𝑖𝑎𝑛 = 29.5 +
10
40
20
𝑀𝑒𝑑𝑖𝑎𝑛 = 29.5 + 5
𝑀𝑒𝑑𝑖𝑎𝑛 = 34.5