The middle value in the list of numbers is called median.If there are two middle numbers the median is the arithmetic mean of the two middle numbers. The median of a set of numbers is the number in the middle.
For example, in the set of numbers {4,6,25}, the median is 6. However, the numbers must be in order for the median to be in the middle. If there are an even number of numbers, then the median is the average of the last 2 middle numbers. There are 2 ways to find the median of a set of numbers:
Rewrite the numbers in order, then find the one in the middle Cross off the highest number, then the lowest, then the highest, lowest, on and on, until only one number is left. That number will be the median.
For example:
1
3
5 \(\leftarrow\) median=5
9
12
Notice that you should arrange the data in ascending order or descending order to find the median.
Examples
Solution:
Arranging the data in ascending order,
8, 9, 10, 12, 12, 13, 14, 16
N= 8
Median (M_{d}) = \(\frac{N + 1}{2}\)
= \(\frac{8+1}{2}\)
= \(\frac{9}{2}\)
= 4.5
Since, the datas are even, the 4^{th} and 5^{th} terms are added to get median.
Median (M_{d}) = \(\frac{Fourth+ Fifth}{2}\)
M_{d} = \(\frac{12 + 12}{2}\)
(Median) = 12
Solution:
Arranging the data in ascending order,
6, 10, 12, 15, 24, 30, 12
There are seven values. The middle value is 15.
So, median= 15.
Solution:
x - 1, x + 2, 2x -1, 3x + 1 and 4x - 1 are in ascending order.
The median = 2x - 1
So, 17 = 2x - 1 ( median = 17)
Or, 2x = 18
Or, x = \(\frac{18}{2}\)
\(\therefore\) x = 9
Solution:
Arranging the data in ascending order, 5, 15, 20, 25, 30, 35.
There are six values. There are two middle numbers 20 and 25.
So, median = \(\frac{20 + 25}{2}\) = \(\frac{45}{2}\) = 22.5
Solution:
Arranging data in ascending order.
15, 20, 25, 30, 35, 40, 45
N = 7
Then, Median lies in (\(\frac{N+1}{2}\))^{th} item
= \(\frac{7+1}{2}\)^{th} item
= \(\frac{8}{2}\)^{th} item
= 4^{th} item
Again 4^{th} item lies in the average of 4th item.
Therefore, median =30.
Solution:
Arranging the data in the ascending order.
14, 16, 22, 26, 30, 32
Now, median lies in (\(\frac{N+1}{2}\))^{th} item
= \(\frac{6 + 1}{2}\)^{th} item
= (\(\frac{7}{2}\))^{th} item
= 3.5^{th} item
Again, 3.5^{th} item is th eaverage of 3^{rd} and 4^{th} item.
Median = \(\frac{22 + 26}{2}\)
= \(\frac{48}{2}\)
Therefore, median is 24.
Solution:
Arranging the data in ascending order.
13, 20, 22, 30, 34, 47, 56, 61.
N = 8
Median lies in (\(\frac{N+1}{2}\))^{th} item
= (\(\frac{8+1}{2}\))^{th} item
= (\(\frac{9}{2}\))^{th} item
= 4.5^{th} item
Again, 4.5^{th} item average is 4^{th} and 5^{th} item.
Median = \(\frac{30+34}{2}\)
= \(\frac{64}{2}\)
Md = 32
Solution:
Arranging the datas in the ascending order.
17, 23, 36, 42, 47
N= 5
Now,
Median lies in (\(\frac{N+1}{2}\))^{th} item
= \(\frac{5+1}{2}\)^{th} item
= (\(\frac{6}{2}\))^{th} item
= 3^{th} item
Therefore,Median = 36
Solution:
Arranging the datas in the ascending order.
17, 23, 36, 42, 47
N= 5
Now,
Median lies in (\(\frac{N+1}{2}\))^{th} item.
= \(\frac{5+1}{2}\)^{th} item
= (\(\frac{6}{2}\))^{th} item
= 3^{th} item
Therefore, median is 36.
Solution:
Arranging the datas in the ascending order.
15cm, 28cm, 32cm, 40cm, 49cm
Number (N)= 5
Now,
Median lies in (\(\frac{N+1}{2}\))^{th} item
= (\(\frac{5+1}{2}\))^{th} item
=\(\frac{6}{2}\)^{th} item
= 3^{th} item
Therefore, Median = 32 cm.
Solution:
Marks Obtained(x) | Number of Students(f) | (c.f) |
18 | 7 | 7 |
20 | 9 | 16 |
22 | 8 | 24 |
25 | 11 | 35 |
29 | 5 | 40 |
30 | 6 | 46 |
32 | 7 | 53 |
N = 53 |
Here, N = 53
Median(M_{d}) = \(\frac{N+1}{2}\)^{th} item
= \(\frac{53+1}{2}\)^{th} item
= \(\frac{54}{2}\)^{th} item
= 27^{th} item
Then,
In cumulative frequency column, the corresponding value of 27 is 35.
Therefore, Median is 25
Solution:
Arranging the given data in ascending order, 15, 17, 18, 20, 24, 25
There are six values. There are two middle numbers 18 and 20.
So, median =\(\frac{18+20}{2}\)
= \(\frac{38}{2}\)
= 19
Solution:
Given, median = 28
a-1, 2a+1, 3a-2, 4a+2, 5a-1 are in ascending order.
The median = 3a-2
So, 28 = 3a-2
Or, 3a = 28 +2
Or, a = \(\frac{30}{3}\)
\(\therefore\) a=10
Solution:
Given, median = 34
16, 20, 2z - 4, 2z, 32 and 40 are in ascending order.
There are to number in the middle
so,median = 2z - 4 + 2z
or, 34 = 4z - 4
or, 4z = 34 - 4
or, z = \(\frac{30}{4}\)
\(\therefore\) z = 7.5
Solution:
Given, median = 15
6, 8, 2a - 1, 10, 15 are in ascending order.
The median = 2a - 1
or, 15 = 2a - 1
or, 2a = 15 + 1
or, a = \(\frac{16}{2}\)
\(\therefore\) a = 8
Find the median of the each of the following data.
6, 18, 10, 12, 16
Find the median of the following data.
35, 5, 30, 25, 20, 15
Find the median:
28, 6, 25, 10, 24, 30, 12
Find the median.
27, 29, 18, 25, 32, 21, 26
Find the median of the following data:
x | 50 | 100 | 150 | 200 | 250 | 300 | 350 |
f | 50 | 22 | 39 | 41 | 38 | 30 | 20 |
Find the median, from the following data:
Marks | 18 | 20 | 22 | 25 | 29 | 30 | 32 |
No of students | 7 | 9 | 8 | 11 | 5 | 6 | 7 |
Find the median , from the following table:
x | 100 | 200 | 300 | 400 | 500 | 600 | 700 |
f | 8 | 9 | 7 | 15 | 22 | 12 | 10 |
Find the median, from the following data.
34, 46, 49, 38, 56, 86, 68, 35
Find the median.
12, 10, 13, 9, 12, 14, 16, 8
Find the median, from the following data.
Marks obtained | 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 |
No of students | 2 | 3 | 6 | 10 | 12 | 13 | 3 | 4 |
Find the median, from the following data.
5.9 ft, 5.2ft, 6.1ft, 7.2ft, 6.5ft, 5.4ft.
Find the median, of the following data.
250, 282, 211, 190, 235, 284, 237, 217, 245, 257, 281
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Anzali
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