# 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.

## Find the median of the following numbers : 10, 15, 50, 45, 60, 90, 25

#### 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

## Median of coffee prices

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

## Median of sugar packets

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

#### 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.

## 1. Find the median of the grouped data as given in the table :

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

#### 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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