Interquartile Range in Statistics

Last Updated : 08 Nov, 2023

Interquartile Range is a fundamental statistical concept that helps us understand the measure of data spread and variability of data within the given dataset. Interquartile Range is also represented as IQR. The Interquartile Range is quite different from calculating the Range. It is different from Range and it is not affected by outliers. Interquartile Range provides the required valuable information about variability within the middle 50% of the data, which is very useful for descriptive statistics and identifying data issues easily.

Interquartile Range is quite different from the range, because it focuses on the middle 50% of the data, whereas the Range considers all the given data points in the provided data.

Table of Content

What does a Quartile mean?
What is Interquartile Range?
Interquartile Range Formula
How to Find the Interquartile Range?
What is Semi Interquartile Range?

What does a Quartile mean?

Quartiles are defined as a statistical measure which divides the given dataset into four equal parts, which have a first quartile (Q₁), second quartile (Q₂₎ and third quartile (Q₃).

Q₁ is defined as the middle number between the smallest number and the median of the data set.
Q₂ is the median of the data.
Q₃ is the middle value between the median and the highest value of the data set.
Here we use the First Quartile (Q₁) and Third Quartile (Q₃) to calculate the Interquartile Range(IQR).
Whereas, Q₁ represents the 25^th percentile of the data, whereas Q₃ represents the 75^th percentile of the data.

What is Interquartile Range?

IQR is used to measure variability by dividing a data set into quartiles. The data is sorted in ascending order and split into 4 equal parts. Q₁, Q₂, Q₃ called first, second and third quartiles in the given data.

Q₁ represents the 25th percentile of the data.
Q₂ represents the 50th percentile of the data.
Q₃ represents the 75th percentile of the data.

Interquartile Range Meaning

Interquartile range is calculated by using the difference between the third Quartile Q₃ and the first Quartile Q₁. Interquartile range is used to calculate the difference between the upper and lower quartiles in the set of give data. Interquartile range is mostlyuseful measure of variability for skewed distributions.IQR is the range between the first and the third quartiles namely Q₁ and Q₃: IQR = Q₃ – Q₁.

Interquartile Range Formula

The formula used to calculate the Interquartile range is:

Interquartile range = Upper Quartile (Q₃)– Lower Quartile(Q₁)

= Q₃ – Q₁

Interquartile Range Calculation

There are some steps to be followed to calculate Interquartile Range:

Step 1: Arrange the given numbers in the data in ascending order.
Step 2: Count the given number of values in the data.
Step 3: If the total count is odd, then the centre value will be as median, otherwise, calculate the average value for two middle values. This will be considered as Q₂value
Step 4: The Calculated Median divides given values into two equal parts as Lower Half and Upper half. these Lower Half and Upper half are determined as Q₁ and Q₃ parts.
Step 5: From Q1 quartile values, we will calculate one median value.
Step 6: From Q3 quartile values, we will calculate another median value
Step 7: Finally, we will subtract the median values of Q₁ and Q₃.
Step 8: The resulting value is the final interquartile range.

How to Find the Interquartile Range?

we will see how to Calculate the Interquartile Range using an example:

Example: Consider the following dataset of exam scores for a class tenth:

77, 85, 92, 64, 78, 95, 82

Find the Interquartile Range of the above data

Solution:

Now to calculate the Interquartile Range steps involved are:

First, we need to arrange in ascending order
Count the given values i.e is 7 ,so count is odd,then median is middle value =82
Next Divide into two halfs ,Lower half and Upper half
Next identify median value in lower half as Q₁ and upper half as Q₃

Interquartile-Range-Calculation

Now,Q₁= 77 and Q₃= 92

⇒ IQR = Q₃– Q₁= 92-77 = 15

What is Semi Interquartile Range?

Semi interquartile range is also known as the quartile deviation. It help in providing the required information about how the data is distributed around a central point(mean). Half of the difference between the first(Q₁) and third quartiles(Q₃) is the semi-interquartile range. It takes half of the time to cover half of the scores. Extreme scores have little impact on the semi-interquartile range. It is useful for skewed distributions to measure the spread of data.Half of the interquartile range is represented by the semi interquartile range.

How to Find Semi Interquartile Range?

The semi interquartile range is calculated by the following steps:-

Step 1: From the given data, find the first quartile Q₁

Step 2: From the given data, find the third quartile Q₃

Step 3: Next Subtract Q₁ from Q₃

Step 4: Divide by half then we get the semi quartile range.

Step 5: Half of the interquartile range is represented by the semi interquartile range.

Note: The formula to calculate the Semi Interquartile Range = 1/2(Q₃ – Q₁) = Interquartile Range/2

Interquartile Range Median

The term “Interquartile Range Median” is essentially the median of the interquartile range and it provides the measure of central tendency for the middle 50% of your data while minimizing the influence of extreme values.

Median and Interquartile Range

The median is the middle most value of the distribution in the given data. The interquartile range (IQR) is the range of values that resides in the middle of the scores. When a distribution is skewed, and the median is used instead of the mean to show a central tendency, the appropriate measure of variability is the Interquartile range.

Q₁ – Lower Quartile Part in the given data.
Q₂ – Median of the given data.
Q₃– Upper Quartile Part in the given data.

Applications of Interquartile Range

The applications of Interquartile range are mentioned below:

Interquartile Range has got its application in various fields, such as finance, healthcare, and quality control for outlier detection.
Like Range,IQR is not sensitive to its extreme values or outliers in the given dataset.
IQR is mostly useful in measure of variability for skewed distributions.
Interquartile Range acts as a useful tool for summarizing the data and comparing the data, particularly when the data is non-normally distributed data.
Interquartile Range is a versatile tool for in predicting data analysis and outlier detection.
IQR can be used to identify outliers in a data set.
IQR gives the central tendency of the data.

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Solved Examples on Interquartile Range

Example 1: You are given a dataset of the ages of students in a classroom:

18, 19, 20, 21, 22, 35, 13, 23,find the Interquartile Range ?

Solution:

Arrange in ascending order:13, 18, 19, 20, 21, 22, 23, 35

Count the given values i.e is 8, so median is average of two numbers median =20+21/2 = 20.5

Lower half is 13, 18, 19, 20

Median of the lower half (Q₁)= (18+ 19) / 2 = 18.5

Upper half is 21, 22, 23, 35

Median of the Upper half (Q₃)= (22+ 23) / 2 = 22.5

Finally,

IQR = Q₃ – Q₁ = 22.5-18.5 = 4

Example 2: Consider a dataset of exam scores for a class:

Scores: 83, 92, 78, 96, 64, 89, 99, find the Interquartile Range?

Solution:

Arrange in ascending order:64 ,78, 83, 89, 92, 96 ,99.

Count the given values i.e is 7, so median is middle most value ,median=89

Lower half is 64 ,78, 83

Median of the lower half (Q₁)= 78

Upper half is 92, 96 ,99

Median of the Upper half (Q₃)= 96

Finally,

IQR = Q₃ – Q₁ =96-78 = 18

Example 3: Imagine a dataset of monthly rainfall (in millimeters) for a city for the past year:

Rainfall: 50, 48, 52, 58, 45, 70, 65, 80, 40 find the IQR of monthly rainfall for the city?

Solution:

Arrange in ascending order:40,45,48,50,52,58,65,70,80

Count the given values i.e is 9, so median is middle most value ,median=52

Lower half is 40,45,48,50

Median of the lower half (Q₁)= 45+48/2 =46.5

Upper half is 58,65,70,80

Median of the Upper half (Q₃)= 65+70/2 =67.5

Finally,

IQR = Q₃ – Q₁ =67.5-46.5 = 21

Example 4: The age of a group of young gymnasts are 4, 5, 6, 3, 12, 14, 15, 13 Find the interquartile range, and the semi-interquartile range?

Solution:

Arrange in ascending order:3,4,5,6,12,13,14,15

Count the given values i.e is 8, so median is average of two numbers median =6+12/2= 9

Lower half is 3,4,5

Median of the lower half (Q₁)= 4

Upper half is 13,14,15

Median of the Upper half (Q₃)= 14

Finally,

IQR = Q₃ – Q₁ =14-4 = 10

Semi Interquartile Range = IQR/2 = 10/2 = 5.

Example 5: Consider a dataset of test scores for a class:

Test Scores: 85, 94, 78, 96, 64, 89, 99, find the Interquartile Range and semi Interquartile Range?

Solution:

Arrange in ascending order:64,78,85,89,94,96,99

Count the given values i.e is 7, so median is middle most value ,median=89

Lower half is 64,78,85

Median of the lower half (Q₁)= 78

Upper half is 94,96,99

Median of the Upper half (Q₃₎=96

Finally,

IQR = Q₃ – Q₁ =96-78 =18

Semi Interquartile Range = IQR/2 =18/2 =9.

Practice Questions On Interquartile Range

Q1. Calculate the Interquartile Range for the following dataset: 12, 15, 20, 25, 30, 35, 40, 45?

Q2. A dataset of temperatures in degrees Celsius for a week is given as follows: 18, 22, 20, 25, 19, 28, 17. Find the Interquartile Range?

Q3. You have a dataset of the heights (in inches) of a group of individuals: 62, 67, 71, 68, 70, 75, 61, 66, 69, 70. Determine the Interquartile Range of heights?