Median formula

 

                

              

MEDIAN

MEDIAN

    Median  is a measure of central tendency, which denotes the values of the 

middle most observation in the data . It is the value that divides a data sample, a 

population, or a probability distributions upper and lower halves in statistics and 

probability theory. There are two types of observations - Ungrouped and grouped. 

Median of  an Ungrouped data:

      The Median of an ungrouped data is the number which is in  the middle when 

they are arranged in ascending or descending order .

       If the number  of variates is odd then [(n+1)/2]th  variate is the median

       If the number of variate is even  then (n/2) th variate is the median

       i.e,  if n is odd , Median = [(n+1)/2]th observation.

              if n is even , Median =(n/2)th variate +  (n+1)/2th variate  

                                                                         2

Example: 

Find the median  

          9,7,6,12,13,5,3

solution:

   Firstly ,arrange the data in ascending or descending order

    Ascending order : 3 , 5 ,6 , 7 ,9 , 12 , 13

  here n=7, which is odd

   Therefore Median = (n+1)/2

                                = (7+1) / 2

                                = 4th variate

                                = 7

    

Example:

Find the median :

     12 , 6 , 17,  8, 15, 34 ,13, 22

solution:

  Firstly, arrange the data in ascending or descending order

  Here the no. of variate or observation is an even number.

  so for an even number , Median = [(n/2)th observation +(n+1)/2 th observation ] /2

                                                          = [8/ 2 th observation +  9/2th observation] / 2

                                                          = [4 th +5th observation] /2

                                                          = [8 +15] /2

                                                            = 23/ 2

                                                           = 11.5 

 

Median of a Grouped data:

Median of a grouped data is the data arranged in ascending order and it is the 

middle most value that separates the higher half of the data from the lower half . 

The data is in the form of a frequency distribution table . To find the median of a 

grouped data we have a specific formula . To find median using that formula we 

have to follow some steps:

        1.  Calculate the total no. of observations .

        2. Write down  the class size and divide the data into different sizes.

        3 . Find the cumulative frequency of each data.

        4. Identify the class in which median falls (n/2 lies).

        5. Find the lower limit of the median class(l) and the cumulative frequency of 

the median class (c).

               median formula  for a grouped data is given by:

                                 

                            Median=l+f2N​−c f​×h

         where l = lower limit of the median class.

                    c f = cumulative frequency of the class preceding the median class.

                    n = no. of observations .

                     f = frequency of the median class.

                     h = class width.

Examples;

Find the median class of the following distribution:

Class	0−10	10−20	20−30	30−40	40−50	50−60	60−70
Frequency	4	4	8	1	1	8	4

solution 

class	frequency	cumulative frequency
0-10	4	4
10-20	4	8
20-30	8	16
30-40	10	26
40-50	12	38
50-60	8	46
60 -70	4	50

here ,

     N/2 = 50/ 2

           =25

     median class = 30 - 40

     f = 10 (frequency of the median class)

     h = 10 (upper limit - lower limit)

     l = 30 ( median class lower limit) 

     c f =  16  (c f of the class preceding the median class)

         

        Median = l + (N/2-cf) x h

                                f

                      = 30 +(25- 16) x 10

                                    10

                      = 30 +9

                      = 39

Example

The median of the following data is 525. Find the values of x and y, if the total frequency is 100

Class interval	Frequency
0−100	$2$
100−200	$5$
200−300	$x$
300−400	$12$
400−500	$17$
500−600	$20$
600−700	y
700−800	$9$
800−900	$7$
900−1000	$4$

solution     

Class interval	Frequency (f)	Cumulative frequency (c f)
0-100	2	2
100-200	5	7
200-300	x	7+x
300-400	12	19+x
400-500	17	36+x
500-600	20	56+x
600-700	y	56+x+y
700-800	9	65+x+y
800-900	7	72+x + y
900-1000	4	76+x + y
		Total = 100

$N=\sum f_i=100$
$\Rightarrow 76+x+y=100$

$i.\; e,\; x+y=24$
It is given that the median is $525$. Clearly, it lies in the class $500-600$ 
$\therefore l=500$

$h=100$

$f=20$

$F=36+x$ 

      $N=100$
   Now,
    Median=l+f2N​−cf​×h
$\Rightarrow 525=500+\frac{50-(36+x)}{20}\times 100$
$\Rightarrow 525-500=(14-x)\times 5$
$25=70-5x$

$5x=45$

$x=9$
     Putting $x=9$ in $x+y=24$, we get $y=15.$
     Hence, $x=9$and $y=15.$

Example

   

A life insurance agent found the following data for the distribution of ages of 100 policy holders. Calculate the median age, if policies are only given to persons having age 18 years on wards but less than 60 years.

Age in years	Number of policy holders
Below 20	$2$
Below 25	$6$
Below 30	$2$
Below 35	$4$
Below 40	$7$
Below 45	$8$
Below 50	$9$
Below 55	$9$
Below 60	$1$

 

  solution

  

Class interval	Frequency	Cumulative frequency
$15-20$	2	2
$20-25$	4	6
$25-30$	1	2
$30-35$	2	4
$35-40$	3	7
$40-45$	1	8
$45-50$	3	9
$50-55$	6	9
$55-60$	2	1

$\Rightarrow$  $N=100$ then, $\frac{N}{2}=50$

The cumulative frequency just greater then $50$ is $78$.

$\therefore$  Median class $=35-40$

      Here, $l=35$

$h=5$

$f=33$

$cf=45$

$Median=l+\frac{\frac{N}{2}-cf}{f}\times h$

  $Median=35+\frac{50-45}{33}\times 5$

  $Median=35+\frac{25}{33}=35.76$

$Example:$

  

Find the median of the following distribution

x	1	2	3	4	5	6	7
f	7	13	12	8	21	9	11

 Solution

    

x	$f$	Cumulative Frequency
1	7	7
2	1	$7+13=20$
3	1	$20+12=32$
4	8	$32+8=40$
5	2	$40+21=61$
6	9	$61+9=70$
7	1	$70+10=81$
Here $N=81$		$\Rightarrow \frac{N}{2}=40.5$

We find that the cumulative frequency greater than $\frac{N}{2}(40.5)$ is $61$ and value of $x$ corresponding to $61$ is $5$. 

Hence, $Median=5$

Example:

 

Calculate the missing frequency from the following distribution, it being given that the median of the distribution is 24.

Age in years	0-10	10-20	20-30	30-40	40-50
No. of persons	$5$	$25$	?	$18$	$7$

solution

 Let the missing frequency be x.

Age in years (x)	No. of persons (f)	C.F.
0-10	5	5
10-20	25	30
20-30	x	30 +x
30-40	18	48 +x
40-50	7	55 +x
	N=55+x

         As, N=55+x 

2N​=255+x​

        Median=l+f2N​+C​×h

         As median is given 24, 

     therefore 

                 median class is 20−30

  

           l=20 (lower limit of the        median class)

          C=30 (C.F. of the class preceding the median class )

      

            h= 40 -30(higher limit - lower limit)

           f=x (frequency of median class )

∴median=20+x255+x​−30​×10

             24=20+2xx−5​×10

                 4=xx−5​×5      

       

                 4x = 5x -25           

 

                        x=25

    

   therefore missing frequency is 25

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