What Does Interquartile Range Mean?
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
Arrange the numbers given in the data in an increasing order
Count the number of values in the data provided to you.
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
Median divides given values into two equal parts. They are determined as Q1 and Q3 parts.
From Q1 values, we will calculate one median value.
From Q3 values, we will calculate another median value.
Finally, we will subtract the median values of Q1 and Q3.
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.
FAQs on Interquartile Range
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 the 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 the interquartile range when we divide it by 2. Hence, it is also known as the 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 the 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 the median.
Third Quartile- It divides the data such that three-fourths or 75% of the observation lies below the third quartile and one-fourth or 25% of the observation lies above the third quartile. The third quartile is also known as the upper quartile. It is represented as Q3.
3. How should you prepare for interquartile range?
The interquartile range is a tricky topic that needs to be well revised. For this take help from the internet, you can use revision notes and you can take help from video lectures of interquartile range present on the internet. Besides this practice as many practice problems as you can and along with this you should solve previous year's question paper and mock test if you want to score well in examinations. In this way, you can get a good grip on your topic.
4. What is the main purpose of interquartile range?
Interquartile range popularly known as IQR is the best measure for variability for the data distribution that is in skewed format or for the data sets that have the outliers. This is because its values come from the middle half of the data distribution which is not easily influenced by the outliers. This range gives us the information of how much the database is spread out. It gives us the inflation of how far apart is both the first quartile and the third quartile.