Median
Median
The median is a variate value which divides the given data into two equal parts. If the data are arranged either in ascending or descending order, the middlemost number is median.
For example,
1, 3, 8, \(\underline{10}\), 13, 16, 20
The middle number 10 is median.
Calculation of median for individual series:
The formula for calculating median for individual series is given by;
Median (Md) = Position of \((\frac{N + 1}{2})\)^{th} item
where, N = No. of observation or items.
The steps for calculation of median for individual series:
 Arrange the data in ascending or descending order.
 Count the number of terms (n).
Median for a discrete data:
Let, x_{1}, x_{2}, x_{3..................}x_{n} be the variable values and f_{1}, f_{2}, f_{3},............ f_{n} be their corresponding frequency respectively.
Then, Median (Md) = Position of \((\frac{N + 1}{2})\)^{th} item
where, N = total sum of a frequency
Median for a continuous data:
To calculate the median for a continuous data, the following steps are applied:
 Prepare cumulative frequency table
 Find the value of \((\frac{N}{2})\)^{th} items which gives the median class.
 Use the formula, Median = l + \(\frac{\frac{N}{2}− c.f}{f}\) × i
where,
l = lower limit of median class
c.f = cumulative frequency
f = frequency
i = class interval
 Calculation of median for individual series = Position of \((\frac{N + 1}{2})\)^{th} item
 Median for a discrete data = Position of \((\frac{N + 1}{2})\)^{th} item
 Median for a continuous data = L + \(\frac{\frac{N}{2}− c.f}{F}\) × i
First, arrange the numbers from least to greatest. Remember that with negative numbers, larger numbers like 9 (if you ignore the minus sign) are less than smaller numbers.
9  8  5  5  5  4  3  0 
There is an even number of numbers, so there are two numbers in the middle.
9  8  5  5  5  4  3  0 
The median is the mean of the two middle numbers. Find the mean of 5 and 5.
10 ÷ 2 = 5
Hence, The median is 5.
First, arrange the numbers from least to greatest:
9.1  9.1  9.3  9.3  9.3  9.5  9.8  
9.8  9.9 
Now find the number in the middle.
9.1  9.1  9.3  9.3  9.3  9.5  9.8  
9.8  9.9 
The number in the middle is 9.3.
Hence, The median voltage was 9.3 volts.
Solution:
Here, Given data is 10 ,14, 16,. 20, 22, 25, 28
Numbers of terms(n) = 7
We have, Median =( \(\frac{n+1}{2}\))^{th }item
( \(\frac{7+1}{2}\))^{th }item
= (\(\frac{8}{2}\))^{th} item
= 4^{th} item
\(\therefore\) The median of the given data is 20.
Solution:
Arranging the data in ascending order
10, 12, 16, 20, 24, 28 , 30, 32
Number of terms(n) = 8
We have,
Median = (\(\frac{n+1}{2}\))^{th} item
= (\(\frac{8+1}{2}\))^{th} item
= (\(\frac{9}{2}\))^{th} item
= (4.5)^{th} item
Now,
Median = mean of 4^{th} item and 5^{th} item
= \(\frac{20+24}{2}\)
= \(\frac{44}{2}\)
= 22
\(\therefore\) Median = 22
Find the median of the given data:
x  10  15  20  25  30 
f  2  4  6  5  4 
Solution:
x  f  c.f 
10  2  2 
15  4  6 
20  6  12 
25  5  17 
30  4  21 
N = 21 
We have,
Median = value of (\(\frac{n+1}{2}\))^{th} item
= value of (\(\frac{21+1}{2}\))^{th} item
= value of 11^{th} item
In c.f. just greater than 11 is 12 and its corresponding value is 20.
\(\therefore\) Median is 20.
Find the median from the given data:
x 
10 
16 
20 
25 
30 
35 
50 
f 
4 
6 
10 
15 
20 
12 
8 
Solution:
x  f  c.f. 
10  4  4 
16  6  10 
20  10  20 
25  15  35 
30  20  55 
35  12  67 
50  8  75 
N=75 
We have, = value of (\(\frac{N+1}{2}\))^{th} item
= value of (\(\frac{75+1}{2}\))^{th} item
= value of 38^{th} item
In c.f. column, c.f. just greater then 38 is 55 and its corresponding value is 30.
\(\therefore\) Median = 30
Find the median from the given data:
Marks 
80 
75 
60 
55 
50 
40 
35 
no. of students 
3 
7 
10 
6 
4 
2 
1 
Solution:
Marks  f  c.f. 
80  3  3 
75  7  10 
60  10  20 
55  6  26 
50  4  30 
40  2  32 
35  1  33 
N=33 
We have, = value of (\(\frac{N+1}{2}\))^{th} item
= value of (\(\frac{33+1}{2}\))^{th} item
= value of 17^{th} item
In c.f. column, c.f. just greater then 17 is 20 and its corresponding value is 60.
\(\therefore\) Median = 60
Find the median from the given data:
height (cm) 
110 
115 
120 
124 
128 
130 
no. of girls 
3 
5 
10 
7 
3 
2 
We have, = value of (\(\frac{N+1}{2}\))^{th} item
= value of (\(\frac{40+1}{2}\))^{th} item
= value of 20.5^{th} item
In c.f. column, c.f. just greater then 20.5 is 28 and its corresponding value is 120.
\(\therefore\) Median = 120
Find the median from the given data:
x 
10 
20 
30 
40 
50 
f 
4 
6 
10 
8 
5 
Solution:
x  f  c.f. 
10  4  4 
20  6  10 
30  10  20 
40  8  28 
50  5  33 
N=33 
We have, = value of (\(\frac{N+1}{2}\))^{th} item
= value of (\(\frac{33+1}{2}\))^{th} item
= value of 17^{th} item
In c.f. column, c.f. just greater then 17 is 20 and its corresponding value is 30.
\(\therefore\) Median = 30
Find the median from the given data:
Age (in yrs)  04  48  812  1216  1620 
No. of students  2  4  8  6  2 
Solution:
Class  Frequency  c.f. 
04  2  2 
48  4  6 
812  8  14 
1216  6  20 
1620  2  22 
N=22 
Here, \(\frac{N}{2}\) = \(\frac{22}{2}\) =11
Median Class = value of (\(\frac{N}{2}\))^{th} item
= (11)^{th} item
= (812)
Here,
l = 8, (\(\frac{N}{2}\))^{th} item = 11, c.f.= 6, f = 8, i= 4
Now,
median = l + \(\frac{\frac{N}{2}− c.f}{f}\) × i
= 8 + \(\frac{11−6}{8}\) ×4
= 8+\(\frac{5}{8}\) ×4
= 8+ 2.5
= 10.5
\(\therefore\) median =10.5
Find the median from the given data:
Marks  510  1015  1520  2025  2530 
Frequency  20  30  50  40  10 
Solution:
Marks  Frequency  c.f. 
510  20  20 
1015  30  50 
1520  50  100 
2025  40  140 
2530  10  150 
N=150 
Here, \(\frac{N}{2}\) = \(\frac{150}{2}\) =75
Median Class = value of (\(\frac{N}{2}\))^{th} item
= (75)^{th} item
= (1520)
l = 15, (\(\frac{N}{2}\))^{th} item = 75, c.f.= 50, f = 50, i= 5
Now,
median = l + \(\frac{\frac{N}{2}− c.f}{f}\) × i
=15 + \(\frac{75−50}{50}\) ×5
= 15+\(\frac{25}{50}\) ×5
= 15+2.5
= 17.5
\(\therefore\) median =17.5
Find the median from the given data:
x  010  1020  2030  3040  4050  5060  6070 
f  3  6  8  10  15  12  6 
Solution:
x  f  c.f. 
010  3  3 
1020  6  9 
2030  8  17 
3040  10  27 
4050  15  42 
5060  12  54 
6070  6  60 
N=60 
Here, \(\frac{N}{2}\) = \(\frac{60}{2}\) =30
Median Class = value of (\(\frac{N}{2}\))^{th} item
= (30)^{th} item
= (4050)
Here,
l = 40, (\(\frac{N}{2}\))^{th} item = 30, c.f.= 27, f =15, i=10
Now,
median = l + \(\frac{\frac{N}{2}− c.f}{f}\) × i
=40 + \(\frac{30−27}{15}\) ×10
= 40+\(\frac{3}{15}\) ×10
= 40+2
= 42
\(\therefore\) median = 42
Find the median from the given data:
x 
010 
1020 
2030 
3040 
4050 
5060 
6070 
f 
5 
8 
11 
15 
20 
12 
9 
Solution:
x  f  c.f. 
010  5  5 
1020  8  13 
2030  11  24 
3040  15  39 
4050  20  59 
5060  12  71 
6070  9  80 
N=80 
Here, \(\frac{N}{2}\) = \(\frac{80}{2}\) =40
Median Class = value of (\(\frac{N}{2}\))^{th}item
= (40)^{th}item
= (4050)
Here,
l= 40, (\(\frac{N}{2}\))^{th}item = 40, c.f.= 39, f =20, i=10
Now,
median = l + \(\frac{\frac{N}{2}− c.f}{f}\) × i
= 40 + \(\frac{40−39}{20}\) ×10
= 40+\(\frac{1}{20}\) ×10
= 40+0.5
= 40.5
\(\therefore\) median =40.5
Find the median from the given data:
class interval 
05 
510 
1015 
1520 
2025 
2530 
frequency 
5 
9 
15 
22 
18 
11 
Solution:
x  f  c.f. 
05  5  5 
510  9  14 
1015  15  29 
1520  22  51 
2025  18  69 
2530  11  80 
N=80 
Here, \(\frac{N}{2}\) = \(\frac{80}{2}\) =40
Median Class = value of (\(\frac{N}{2}\))^{th}item
= (40)^{th}item
= (1520)
Here,
l= 15, (\(\frac{N}{2}\))^{th}item = 40, c.f.= 29, f =22, i=5
Now,
median = l + \(\frac{\frac{N}{2}− c.f}{f}\) × i
= 40 + \(\frac{40−29}{22}\) ×5
= 40+\(\frac{11}{22}\) ×5
=40+2.5
=42.5
\(\therefore\) median = 42.5

In the data set below, what is the median?
1, 2. 4, 4, 5, 6, 6, 6 , 7, 7, 8
4
7
6
5

In the data set below, what is the median?
2, 2, 2, 2, 3 , 5, 5, 5, 7, 8, 9, 95
7
8
3

In the data set below, what is the median?
19 37 41 43 83 85 94
37
43
41
83

In the data set below, what is the median?
2, 4, 5, 6, 6, 7, 8, 8, 8, 9
8
5
7
6

Find the median.
64, 60, 70, 72, 68, 80, 85, 56
68
69
75
41

Find the median.
47, 61, 13, 34, 56, 22, 8
54
34
56
45

Find the median.
8, 1, 3, 3, 1, 7, 4, 1, 4, 4
8
32
45
3.5

x1, 2x+1, x+5 and 3x+1are ascending order. If its median is18, find the value of x.
3
45
12
1

12, 17, 2x+3, 3x+5, 36, 43 are in ascending order. If its median is 29, find the value of x.
10
12
15
20

6, 8, 10, 12, 2x, 2x+2, 18, 20, 22, 24 are in ascending order. If its median is 15, find the value of x.
8
5
12
7

Find median. 12, 1, 10, 1, 9, 3, 4, 9, 7, 9
2
5
8
1

Find median.
85, 77, 60, 89, 91, 89, 78
60
45
85
47

What is value of x if its median is 12. x+1, 2x1, x+7 and 3x+4
4
13
6
2

Find median. 15, 10, 11, 9, 14
20
15
11
12

Find median. 14, 16, 12, 6, 8,10
4
15
12
11

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