 # Interquartile Range

## What Does Interquartile Range Mean?

The interquartile range is a measure of variability based on splitting data into quartiles.

Quartile divides the range of data into four equal parts. The values that split each part are known as the first, second and third quartile. And they are represented by Q₁, Q₂, and Q₃.

Q₁- It is the middle value in the first half of the rank-order data

Q₂- It is the median value in the set

Q₃- It is the middle value in the second half of the rank-ordered data set.

Interquartile range is equal to quartile 3 minus quartile 1.

### Interquartile range Definition and Example

The interquartile definition states that the interquartile range is the difference between the third and the first quartiles. As we know quartiles are the divided values that divide the complete series into four equal parts. So, there are total of 3 quartiles The first quartile also known as the lower quartile is represented by Q₁, the second quartile is represented by Q₂, and the last third quartile also known as upper quartile is represented by Q₃.

Let's calculate the interquartile range of the below data

## Inter-Quartile Example on The Basis of The Above Definition

 Lower Half or First Quartile   62    63    64   64   70 Median   OrSecond quartile Upper Half or Third Quartile72   76   77   81   81

In the above data, Q₁ part is 62, 63, 64, 64, 70

Q₁ =  As there are 5 values in the lower half, so the Q₁ will be 64 as it is a middle value of lower half

There are a  total of 10 values in above data

As, we know 10 is an even number so the median is mean of 70 and 72

Hence, the Q₂ is 71

Q₃ part is  76,77,81, 81

Q₃ = As there are 5 values in the upper half, so the Q₃ will be 77 as it is a middle value of the upper half.

Interquartile range = Q₃ - Q₁ = 77 -64 = 13

Hence, the quartile range of the above data is 13.

### Interquartile Range Formula

Interquartile range is the difference between the upper quartile and the lower quartile. The interquartile range formula is given below.

Interquartile range - Upper quartile- Lower quartile = Q₃ - Q₁

In the above interquartile range formula, Q₁ represents the lower quartile whereas Q₃ represents the upper quartile.

How to Calculate The Interquartile Range?

Here are some of the steps to calculate the interquartile range

1. Arrange the numbers given in the data in an increasing order

2. Count the number of values in the data provided to you.

3. If the total number of values is odd, then the center value will be considered as median otherwise calculate the mean value for two middle values. This will be considered as Q₂ value

4. Median divides given values into two equal parts. They are determined as Q1 and Q3 parts.

5. From Q1 values, we will calculate one median value.

6. From Q3 values, we will calculate another median value.

7. Finally, we will subtract the median values of Q1 and Q3.

8. The resulting value will be the interquartile range of the given data

### Interquartile Range Example

The following interquartile range example helps you to understand the concept of interquartile range thoroughly.

1. Calculate the Interquartile Range for The Following Data

 4 17 7 14 8 12 3 16 10 4 4 1

Solution:

Arrange the above data in ascending order

 3 4 4 4 7 10 11 12 14 16 17 18

Divide the above data into 4 quarters

Numbers lie in quarter 1 -  3, 4,  4

Numbers lie in quarter 2-   4,  7,  10

Numbers lie in quarter 3-  11, 12, 14

Numbers lie in quarter 4 -  6, 17, 18

Q₁ = (4+4)/2 = 4

Q₂  = (10+ 11)/2 = 10.5

Q₃  = (14 + 16)/2 = 15

Interquartile range = Q₃ - Q₁ = 15-4

= 11

Hence, the interquartile range for the above interquartile range example is 11

### Why Is Interquartile Range Important?

Besides being a less sensitive measure for the spread of the data, the interquartile range is extremely important. As the interquartile range is resistant to outliers, the interquartile range is used to identify when a value is in outliners. It tells us how spreaded our entire data is

Through interquartile range, we get to know whether the outlier we have is mild or strong. To look for an outlier, we should look either below the first quartile or above the third quartile. How far we should proceed relies on the values of the interquartile range.

The main use of interquartile range in place of range for the measurement of the spread of data is that interquartile range is not sensitive to outliers.

Solved Examples

1. Find the interquartile range for the first ten prime numbers

Solutions: The first 10 prime numbers are :

 2 3 5 7 11 13 17 19 23 29

As, the above data is already in ascending order, so we don't need to arrange it.

Q₁ part is-  2,3,  5, 7, 11

The number of values in Q₁ is 5

5 is an odd number, hence the middle value is 5, i.e Q₁ is 5

There are a total of 10 values in the above data

As we know 10 is an even number so the median is mean of 11 and 13

Hence, the Q₂ is 12

Q₃ part is - 13,17,19,23,29]

The number of values in Q₁ is 5

5 is an odd number, hence the middle value is 19, i.e Q₃ is 19

Q₁ = (4+4)/2 = 4

Q₂  = (10+ 11)/2 = 10.5

Q₃  = (14 + 16)/2 = 15

Interquartile range = Q₃ - Q₁ = 19-5

= 11

Hence, the interquartile range for the above data is 11

1. ### Find the interquartile range for the below data

 56 14 84 21 85 2 35 74 66 52 45

Solution: The first step is to arrange the data in ascending order

 2 14 21 35 45 52 56 66 74 84 85

Q₁ = (N + 1)/ 4 term

= (1 1+ 1) /4 term

= 3rd term

And Q₃ = 3 x (N + 1)/ 4 term

= 3 (11 + 1)/4 term

= 3 x 3 term

= 9th term

Interquartile range  = Q₃ - Q₁ = 74 - 21 = 53

1. Explain The  median and the interquartile range?

The median is the middle value of the distribution of the given data. The interquartile range is the range of values that lies in the middle of the scores.The median is used in place of mean to determine central tendency if the distribution is skewed. The most appropriate measure of variability is the interquartile range.

Q₁- Lower Quartile portion

Q₂- Median

Q₃- Upper quartile portion

The interquartile range is the measure of central tendency on the basis of the lower and upper quartile. We get quartile deviation from interquartile range when we divide it by 2.Hence  it is also known as semi- interquartile range.

2. Explain the Quartiles?

The quartile divides the series of data into four equal parts. The four parts of the quartile are First Quartile, Second Quartile, Third Quartile, and Fourth Quartile. The Second Quartile is also known as the median of the data series as it divides the data into equal parts.

First Quartile- It divides the data such that 25% of value lies below the first quartile and the remaining 75% of the values lies above the first quartile. The first quartile is also known as the lower quartile. It is represented as Q1

Second Quartile- It divides the data into two equal halves such that 50% of the observations lie below the second quartile and another 50% of the observation lies above the second quartile. The second quartile is also known as medan. It is represented as

Third Quartile- It divides the data such that three-fourth or 75% of the observation lies below the third quartile and one-fourth or 25% of the observation lies above the thirds quartile. The third quartile is also known as the upper quartile. It is represented as Q3.