[Maths Class Notes] on Interquartile Range Pdf for Exam

The interquartile range is actually the measure of static depression and the spread of the data. Instead of data, there are two quartiles, namely the upper quartile and the lower quartile. The interquartile range is used to explain the difference between the upper and lower quartiles in the set of data. The interquartile range can be used to denote or indicate the variability of the set provided to you.

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. 

The interquartile range is equal to quartile 3 minus quartile 1.

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Outliers 

The interquartile range is often used to measure or find the outliers in the data. From the data or on a box plot a fence is used to identify and categorize the type of outliers. If we talk about fences then there are four relevant fences. Let’s have a look at them:

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Lower inner fence Q₁– 1.5 * IQR 

Upper inner fence Q₃ +1.5 * IQR 

Lower outer fence Q₁– 3 * IQR 

Upper outer fence Q₃+ 3* IQR 

Q₁ here is the middle value in the first half

Q₂ is the median value in the set.

Q₃ is the middle value in the second half.

Any data point that is going to fall between the inner and outer fences is said to be the mid-outliners. And all the points that fall beyond the outer fences are known as extreme outliers.

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 a 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 the 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

Table

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 the lower half

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 70 and 72

Hence, the Q₂ is 71

Q₃ part is  72, 76,77,81, 81

Q₃ = As there are 5 values in the upper half, 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

The 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.

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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 centre 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

Solution:

Arrange the above data in ascending order

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

8Q₃  = (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 to spread 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 :

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

Q₁ part is-  2,3,  5, 7, 11The 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

Interquartile range = Q₃ – Q₁ = 19-5= 14

Hence, the interquartile range for the above data is 14

2. Find the interquartile range for the below data

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

Q₁ = (N + 1)/ 4 term

= (11+ 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

Conclusion

This is how we calculate the interquartile range and use the determined formula to calculate the desired outcome of a problem. Learn how the formulas are being used in the solve examples to develop your conceptual foundation.