Quartiles: In statistics, we generally deal with a large amount of numerical data. We have various concepts and formulas in statistics to evaluate the large data. One of such best applications is quartiles. 

Quartiles are the values that divide the given data set into four parts, making three points. Thus, quartiles are the values that divide the given data into three quarters. The middle part of the quartiles tells the central point of distribution. The difference between the higher and lower quartiles so formed gives the interquartile range.

Quartiles Definition

Quartiles are one of the applications of statistics used to evaluate a large number of numerical data. Quartiles are the statistical term that describes the division of a given set of values into four parts, making three points based on the given values of the data.

So, quartile divides the data into four parts, and each part occupies \(\frac{1}{4}^{\text {th }}\) of the data.

Thus, quartiles are the values that divide the given data into four quarters. The three quartiles are first, middle, and the last quartile and they are represented by \(Q_{1}, Q_{2}\) and \(Q_{3}\). Here,

1. \(Q_{1} -\) Lower quartile
2. \(Q_{2} -\) Middle quartile
3. \(Q_{3} -\) Upper quartile

Quartile \(Q_{2}\) gives the median values. The quartile region lies between the lower quartile, and the higher quartile is called an interquartile region.

Learn About Measures of Dispersion

Difference Between Quartile and Quarter

There is a difference between the quartile and quarter. The quarter is the part formed when a whole data is divided into four parts. And the line that is used to divide the data into quarters is called the quartile. Let us understand the above difference by taking a simple example:

When a whole pizza is cut into four parts, each part is the quarter of the pizza, and the line or boundary where pizza is cut into parts is called the quartile.

Quartiles in Statistics

It is important to know about the medians, the middle value of the given set of numbers to understand the quartiles. Median divides the given data into two halves. \(50 \%\) of the data lies above it, and the other \(50 \%\) of the data lies below it. Now, quartiles divide the data into quarters, such as \(25 \%\) of the data lies below the lower quartile, \(25 \%\) of the data lies above the lower quartile, and below the median, \(25 \%\) of the data lies above the median, and below the higher quartile, \(25 \%\) of data lies above the higher quartile.

1. First Quarter: Lies between the lowest value of the data to the lower quartile \(\left(Q_{1}\right)\).
2. Second Quarter: Lies between lower quartile \(\left(Q_{1}\right)\) and middle quartile \(\left(Q_{2}\right)\).
3. Third Quarter: Lies between middle quartile \(\left(Q_{2}\right)\) and higher quartile \(\left(Q_{3}\right)\).
4. Fourth-quarter: Lies between higher quartile \(\left(Q_{3}\right)\) and highest value of the data.

Note: To find the quartiles of any numerical data, the given data should first be arranged in ascending order, i.e. from lowest to highest.

Quartiles Formula

The quartiles formula is used to divide a given set of numbers into quarters. There are three quartiles formed, dividing the given data into four quarters. We are arranging the data in ascending order. The first quartile lies between the first number and the median, the second quartile is the median, and the third quartile lies between the median and the last number.

When the set of \(n\) numbers are arranged in ascending order, then the formula used to calculate the quartiles are given below:

1. First quartile \(\left(Q_{1}\right)=\left(\frac{n+1}{4}\right)^{\text {th}}\) term
2. Second quartile \(\left(Q_{2}\right)=\left(\frac{n+1}{2}\right)^{\text {th}}\) term
3. Third quartile \(\left(Q_{3}\right)=\left(\frac{3(n+1)}{4}\right)^{\text {th}}\) term

Example: Find the quartiles for the given data: \(4, 5, 6, 2, 4, 8, 7\).
Arranging the data in ascending order as follows: \(2, 4, 4, 5, 6, 7, 8\)

1. Median is \(\frac{(7+1)^{\text {th}}}{2}=4^{\text {th }}\) term \(=5\), which is the second quartile.
2. First quartile \(=\frac{(7+1)^{\text {th}}}{4}=2^{\text {nd}}\) term \(=4\).
3. Third quartile \(=\frac{3(7+1)^{\text {th}}}{4}=6^{\text {th}}\) term \(=7\).

Steps for Finding Quartiles

The steps to be followed to divide the given data into the quartiles are as given below:

1. Arrange the given numerical data in ascending order.
2. Use the below formulas to find the quartiles:

Interquartile Range

The interquartile range is the distance between the first quartile and the last quartile. Thus, the interquartile range occupies the second \(25\%\) of the data and the third \(25\%\) of the data, divided into four quarters by quartiles.

The interquartile data is generally called IQR. Therefore, IQR measures the middle \(50\%\) of the data. The second quartile in IQR gives the median of the data.

Formula of Interquartile Range

The interquartile (IQR) formula is used to measure the middle \(50\%\) of the data. The interquartile (IQR) is mainly used to measure the variability in the given data set in statistics.

The formula for interquartile (IQR) is given by the difference between the upper or highest quartile (third quartile) and lower or lowest quartile (first quartile).

\(I Q R=Q_{3}-Q_{1}\)

Here, \(\left(Q_{3}\right)=3\left(\frac{n+1}{4}\right)^{\text {th}}\) term and \(\left(Q_{1}\right)=\left(\frac{n+1}{4}\right)^{\text {th}}\) term.

Steps to Find the Interquartile Range (IQR)

The interquartile range is the difference between the upper or highest quartile (third quartile) and the lower or lowest quartile (first quartile). The steps of finding the interquartile range are given below:

1. First, arrange the given data values in ascending order, i.e. from the lowest to the highest. Find the median of the data. Find the second quartile \(\left(Q_{2}\right)\). The median value of the given data can be found based on the number of terms in the data. If the number of values in the given data is odd, then the median \(=\left(\frac{n+1}{2}\right)^{\text {th }}\) term.
2. If the number of values in the given data is even, then the median \(=\frac{\left[\left(\frac{n}{2}\right)^{\text {th}} \text { term }+\left(\frac{n+1}{2}\right)^{\text {th}} \text { term }\right.}{2}\)
3. Find the lower quartile or first quartile \(\left(Q_{1}\right)\). It is the median of the values that lie to the left of the median or second quartile \(\left(Q_{2}\right)\) found in the second step.
4. Find the upper quartile or third quartile \(\left(Q_{3}\right)\). It is the median of the values that lie to the right of the median or second quartile \(\left(Q_{2}\right)\) found in the second step.
5. Now, find the difference between the third quartile and the first quartile, which gives the interquartile range of the given data. \(I Q R=Q_{3}-Q_{1}\)

To represent the IQR, we use the box and whisker plot, which describes that a box is drawn from the lower quartile to upper quartile, which gives the interquartile range (IQR)

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Solved Examples – Quartiles

Q.1. Find the quartiles in the given data \(4, 7, 5, 5, 2, 7, 8\).
Ans:
Arrange the given data in ascending order as follows: \(2, 4, 5, 5, 7, 7, 8\).
1. Median is \(\frac{(7+1)^{t h}}{2}=4^{\text {th}}\) term \(=5\), which is the second quartile.
2. First quartile \(=\frac{(7+1)^{\text {th}}}{4}=2^{\text {nd}}\) term \(=4\).
3. Third quartile \(=\frac{3(7+1)^{\text {th}}}{4}=6^{\text {th}}\) term \(=7\).

Q.2. Find the interquartile range for the given data of the marks secured by the students in the exam is \(62, 70, 72, 63, 64, 81, 77, 81, 76, 64\).
Ans:
The marks of the students are \(62, 70, 72, 63, 64, 81, 77, 81, 76, 64\)
Arranging the data in ascending order: \(62, 63, 64, 64, 70, 72, 76, 77, 81, 81\)
Here, the number of terms is even, then median \( = \frac{{\left[ {{{\left( {\frac{n}{2}} \right)}^{{\rm{th}}}}{\rm{term}} + {{\left( {\frac{{n + 1}}{2}} \right)}^{{\rm{th}}}}{\rm{term}}} \right]}}{2} = \frac{{70 + 72}}{2} = 71.\)
Lower quartile \(\left(Q_{1}\right)=\) Median of the values below the value \(71=64\) and the upper quartile \(\left(Q_{3}\right)=\) Median of the values above the value \(71=77\).
Then, interquartile range \(=Q_{3}-Q_{1}=77-64=13\).

Q.3. Find the upper quartile for the given set of numbers \(26, 19, 5, 7, 6, 9, 16, 12, 18, 2, 1\).
Ans:
Given numbers are \(26, 19, 5, 7, 6, 9, 16, 12, 18, 2, 1\).
Arrange the given data in ascending order: \(1, 2, 5, 6, 7, 9, 12, 16, 18, 19, 26\)
Here, the number of values given, \(n=11\).
The upper quartile is given by \(\left(\frac{3(n+1)}{4}\right)^{\text {th}}\) term \(=\frac{3(11+1)}{4}^{\text {th}}\) term \(=9^{\text {th}}\) term \(=18\).
Hence, \(18\) is the upper quartile for the given set of numbers.

Q.4. Find the interquartile range for the first ten odd numbers.
Ans:
The first ten odd numbers are \(1, 3, 5, 7, 9, 11, 13, 15, 17, 19\).
1. Here, the given numbers are arranged in ascending order.
2. The total number is \(10\), so the median is given by \(\frac{\left[\left(\frac{n}{2}\right)^{\text {th }} \text { term }+\left(\frac{n+1}{2}\right)^{\text {th }} \text { term }\right]}{2}\)
Median \(=Q_{2}=\frac{9+11}{2}=10\).
3. The first quartile is the median value that lies below the median value of \(10\).
The median of \(1,3,5,7,9\) is \(5\).
Thus, first quartile \(=Q_{1}=5\).
4. The third quartile is the median of the value that lies above the median value of \(10\).
The median of \(11,13,15,17,19\) is \(15\).
Thus, third quartile \(=Q_{3}=15\).
5. The interquartile range is \(Q_{3}-Q_{1}=15-5=10\).

Q.5.Find the lower quartile value for the given numbers: \(99, 98, 100, 95, 96, 92, 101,  91, 97, 93, 102\).
Ans:
Given numbers are \(99, 98, 100, 95, 96, 92, 101,  91, 97, 93, 102\).
Arrange the given numbers in ascending order \(91, 92, 93, 95, 96, 97, 98, 99, 100, 101, 102\).
Here, the number of values given, \(n=11\).
The lower quartile is given by \(\left(\frac{(n+1)}{4}\right)^{\text {th}}\) term \(=\frac{(11+1)^{\text {th}}}{4}\) term \(=3^{\text {rd}}\) term \(=93\).
Hence, \(93\) is the lower quartile for the given set of numbers.

Summary

In this article, we have studied the definitions of quartiles, formulas used for the quartiles in detail. We also studied the difference between quartile and quarters. This article gives the definitions of interquartile range and its formula. 

This article also gives the method of finding interquartile range. This article gives the solved examples, which help us t understand the problems easily.

Frequently Asked Questions (FAQs)- Quartiles

Q.1. What is the interquartile range?
Ans: The interquartile range is the distance between the first quartile and the last quartile.

Q.2. What is a quartile?
Ans: Quartiles is the statistical term that describes the division of a given set of values into four parts, making three points based on the given values of the data.

Q.3. How to find the quartiles for the given data?
Ans: When the set of \(n\) numbers are arranged in ascending order, then the formula used to calculate the quartiles are given below:
1. First quartile \(\left(Q_{1}\right)=\left(\frac{n+1}{4}\right)^{\text {th }}\) term
2. Second quartile \(\left(Q_{2}\right)=\left(\frac{n+1}{2}\right)^{\text {th}}\) term
3. Third quartile \(\left(Q_{3}\right)=\left(\frac{3(n+1)}{4}\right)^{\text {th}}\) term

Q.4. How is the interquartile range calculated?
Ans: The formula for interquartile (IQR) is given by the difference between the upper or highest quartile (Third quartile) and lower or lowest quartile (First quartile).
\(I Q R=Q_{3}-Q_{1}\)
Here, \(\left(Q_{3}\right)=3\left(\frac{n+1}{4}\right)^{\text {th}}\) term and \(\left(Q_{1}\right)=\left(\frac{n+1}{4}\right)^{\text {th}}\) term.

Q.5. What is the formula to find the second quartile of the data?
Ans: The second quartile of the data is the median of the data. If the number of values in the given data is odd, then the median \(=\left(\frac{n+1}{2}\right)^{\text {th }}\) term. If the number of values in the given data is even, then the median \(=\frac{\left[\left(\frac{n}{2}\right)^{\text {th}} \text { term }+\left(\frac{n+1}{2}\right)^{\text {th}} \text { term }\right]}{2}\).

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