Note on Median

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Median

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

  1. Find the median of the following data:
    28, 6,2 5, 10, 24, 30, 12
    Solution:
    Arranging the data in ascending order
    6, 10, 12, 15, 24, 28, 30
    There are seven values. The middle value is 15.
    So, the median is 15.

  2. Find the median of the following data:
    35, 5, 30, 25, 20, 15
    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

  • Median is the middle value in the list of numbers.
  • To find median, place the numbers you are given in the  value order and find the middle number.
.

Very Short Questions

Solution:

Arranging the data in ascending order,

8, 9, 10, 12, 12, 13, 14, 16

N= 8

Median (Md) = \(\frac{N + 1}{2}\)

= \(\frac{8+1}{2}\)

= \(\frac{9}{2}\)

= 4.5

Since, the datas are even, the 4th and 5th terms are added to get median.

Median (Md) = \(\frac{Fourth+ Fifth}{2}\)

Md = \(\frac{12 + 12}{2}\)

(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

= 4th item

Again 4th 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

= 3.5th item

Again, 3.5th item is th eaverage of 3rd and 4th 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

= 4.5th item

Again, 4.5th item average is 4th and 5th 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

= 3th 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

= 3th 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

= 3th 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(Md) = \(\frac{N+1}{2}\)th item

= \(\frac{53+1}{2}\)th item

= \(\frac{54}{2}\)th item

= 27th 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

0%
  • Find the median of the each of the following data.

    6, 18, 10, 12, 16

    17
    15
    12
    14
  • Find the median. 12, 34, 21, 18, 23, 30, 27

    25
    33
    23
    32
  • If x - 1, x + 2,2x - 1, 3x + 1 and 4x - 1 are in ascending order having median 17, Find the value of x.

    13
    11
    7
    9
  • Find the median of the following data.

    35, 5, 30, 25, 20, 15

     

    22.5
    27
    24
    29
  • Find the median:

    28, 6, 25, 10, 24, 30, 12

    22
    10
    12
    15
  • Find the median.

    27, 29, 18, 25, 32, 21, 26

    25
    20
    26
    28
  • Find the median of the following data:

     x  50  100  150  200  250  300  350
     f  50  22  39  41  38  30  20

     255
     250
     205
     200
  •  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

     25
     27
     33
     29
  • Find the median , from the following table:

     x  100  200  300  400  500  600  700
     f  8  9  7  15  22  12  10

     650
     550
     500
     555
  • Find the median, from the following data.

    34, 46, 49, 38, 56, 86, 68, 35

     47.5
     55
     48
     50
  •  Find the median.

    12, 10, 13, 9, 12, 14, 16, 8

     15
     12
     22
     17
  • 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

     50
     44
     45
     47
  •  Find the median, from the following data.

     5.9 ft, 5.2ft, 6.1ft, 7.2ft, 6.5ft, 5.4ft.

     4.6
    5.5
    6.0
    5.0
  • Find the median, of the following data.

     250, 282, 211, 190, 235, 284, 237, 217, 245, 257, 281

     255
     245
     267
     250
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X-1, 2x 1, x-5, and 3x 1 are well arranged in proper order. If the median is 18, find the first term


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