Interquartile Range Formula

Interquartile Range is related to the quartile concept which comes under parent topic statistics which is a study of the collection of data, analyzing, interpreting, presenting the data in an organized manner. The interquartile range finds the difference between the upper quartile and lower quartile.

Interquartile Range

The interquartile range can be defined as the distance between the upper quartile and the lower quartile. It is also known as Mid Spread. A quartile is used to partition the given data into four parts by three cuts- Quartile 1, Quartile 2, Quartile 3.

Quartile 1: It is also known as the lower quartile which cuts the first half of sorted data.

Quartile 2: It is a median value which is the centre value in sorted data.

Quartile 3: It is an upper quartile that cuts the last part of the ordered data.

Interquartile Range Formula

Interquartile Range Formula is the difference between the Third/Upper quartile and First/Lower quartile. It is a calculation of variation while dividing the dataset into quartiles. The formula is given by-

Interquartile Range = Q₃ - Q₁

Where 

Q₃-Upper Quartile

Q₁-Lower Quartile

Upper Quartile is calculated using the formula:

Q₃ = ((3 × (n + 1))/4)^th term

where,

n is the number of terms.

Lower Quartile is calculated using the below formula-

Q₁ = ((n + 1)/4)^th term

where,

n is the total number of terms.

Steps to Solve

Step 1: Sort the given data in ascending order.

Step 2: Find Upper Quartile and lower quartile for the given data.

Step 3: Find Inter Quartile Range.

Let's look into a few examples to find the IQR (Inter Quartile Range)

Sample Questions

Question 1: Find Inter Quartile Range for the data 20,10,50,40,25,70,30

Solution:

Step 1: Given data is in unsorted manner. So sort it in ascending order.

10,20,25,30,40,50,70

Step 2: Find first Quartile

Q₁ = ((n+1)/4)^th term

Here n = 7 (Total 7 terms)

    = ((7+1)/4)^th term

    = (8/4)^th term

    = 2^nd term

2nd term is 20

So Quartile₁ = 20

Find Upper/third Quartile

Q₃ = ((3x(n+1))/4)^th term

Here n = 7 (Total 7 terms)

   = ((3×(7+1))/4)^th term

   = ((3×8)/4)^th term

   = (24/4)^th term

   = 6^th term

6^th term is 50

So Quartile₃ = 50

Step 3: Find IQR (Inter Quartile Range)

IQR = Q₃ - Q₁

       = 50 - 20

       = 30

Interquartile Range for the given data is 30.

Question 2: Find Inter Quartile Range for the data 22,12,55,45,25,75,30,26,49

Solution:

Step 1: Given data is in unsorted manner. So sort it in ascending order.

12,22,25,26,30,45,49,55,75

Step 2: Find first Quartile

Q₁ = ((n+1)/4)^th term

Here n = 9 (Total 9 terms)

   = ((9+1)/4)^th term

   = (10/4)^th term

   = 2.5^th term

2.5^th term is average of 2^nd and 3^rd terms 

2.5^th term = (22+25)/2

                = 47/2 => 23.5

So Quartile₁ = 23.5

Find Upper/third Quartile

Q₃ = ((3x(n+1))/4)^th term

Here n = 9 (Total 9 terms)

    = ((3x(9+1))/4)^th term

    = ((3x10)/4)^th term

    = (30/4)^th term

    = 7.5^th term

7.5^th term is average of 7^th and 8^th terms

7.5^th term = (49+55)/2

                 = 104/2

                 = 52

So Quartile₃ = 52

Step 3: Find IQR (Inter Quartile Range)

IQR = Q₃ - Q₁

       = 52 - 23.5

       = 28.5

Interquartile Range for the given data is 28.5.

Question 3: What is the interquartile range for the data if Quartile-1 is 20 and Quartile-3 is 40.

Solution:

Given 

Quartile-3 (Q₃) = 40

Quartile-1 (Q₁) = 20

Interquartile Range = 40 - 20

                               = 20

So interquartile range is 20.

Question 4: What is the interquartile range for the data if Quartile-1 is 10 and Quartile-3 is 90.

Solution:

Given

Quartile-3 (Q3) = 90

Quartile-1 (Q1) = 10

Interquartile Range = 90 - 10

                               = 80

So interquartile range is 80.

Question 5: What is the interquartile range for the data if Quartile-1 is 30 and Quartile-3 is 120.

Solution:

Given

Quartile-3 (Q3) = 120

Quartile-1 (Q1) = 30

Interquartile Range = 120 - 30

                               = 90

So interquartile range is 90.