Median

Median is the "middle" or mid value of a data set.

To find the median, you have to first order the numbers - from smallest to largest, and then find the middle number.

Basic examples


  •   In a set of three numbers - the middle number is the number in 2nd position.

    So, in a set of 10, 20, 30, the middle number is 20.


  •   In a set of five numbers - the middle number is the number in 3rd position since there are two numbers before it and two numbers after it.

    So, in a set of 10, 20, 30, 40, 50, the middle number is 30.


Example with detailed explanation


  45

Explanation:

1. When you put the numbers in numerical order, it will look like this:

10, 15, 25, 45, 50, 60,90


2. There are seven numbers in the series, and out of which the middle number is the fourth from the start. This divides the set into two equal parts - three on the left side and three on the right side of the fourth number.


3. So, the median is 45.


Median of Ungrouped Data


The formulas for calculating the median of an ungrouped data, which has "n" number of observations are:


  •  If the number of observations is odd:


    `x_m` = `x_((n+1)/2)`

  •  If the number of observations is even:


    `x_m` = `(x_(n/2)+x_((n+2)/2))/2`

Ungrouped Data Median Example


Example 1:
There are 7 boxes of coffee, which are priced as below (in US dollars). Find the median.
65, 49, 87, 54, 90, 95, 70.
 70

Explanation :

First of all, we need to arrange the data in the ascending order. The arranged data is:

  •  49
  •  54
  •  65
  •  70
  •  87
  •  90
  •  95

Number of observations = 7 (odd)
Since the number of observations is odd, we can find the median using the following formula:


`x_m` = `x_((n+1)/2)`


`bar(x)` = `(( 7+1 )/2)th` observation = `4^(th)` observation = 70



Example 2:
There are 10 packets of biscuits with sugar content (in grams) as shown below. Find the median.
20, 35, 18, 40, 34 , 42, 25, 30, 29, 44
  32

Explanation :

Given data needs to be arranged in ascending order as shown in the table below :

  • 18
  • 20
  • 25
  • 29
  • 30
  • 34
  • 35
  • 40
  • 42
  • 44

Number of observations = 10 (even)

Since the number of observations is even, we can find the median using the following formula:


`x_m` = `(x_(n/2)+x_((n+2)/2))/2`


`bar(x)= 1/2 [ n/( 2 ) th` observation+`( n/( 2 ) +1)th` observation `]`


` = 1/2 [ 10/( 2 ) th` observation+`( 10/( 2 ) +1)th` observation `]`


`bar(x)= 1/2 [ 5th` observation+`6th` observation `]`


`bar(x)= 1/2 [30+34 ] = ( 64 )/2 = 32`


Median of Grouped Data


To calculate median of a grouped data (which has class and frequency distribution) we need to,

  • 1. Make a new column for cumulative frequency
  • 2. Use the formula `(n/2)^(th)`, where n is the sum of all frequency values.

Grouped Data Median Example

Example:
Number on the Dice (X) Frequency (f)
1 4
2 6
3 2
4 2
5 5
6 1

  2

Explanation:

Make a new column of cumulative frequency by multiplying frequency with class.


Number on the Dice (X) Frequency (f) Cumulative Frequency
1 4 4
2 6 10
3 2 12
4 2 14
5 5 19
6 1 20

Number of observation (n) = 20


Median = class containing `(n/2)^(th)` observation


Median = class containing `(20/2)^(th)` observation


Median = class containing 2nd observation = 2


That is, median of given grouped data is 2.







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