Quartiles: Definition, Formulas, Inter Quartile Range, Examples (2023)

  • Written By Keerthi Kulkarni
  • Last Modified 14-11-2022

Quartiles: Definition, Formulas, Inter Quartile Range, Examples (1)

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.

Quartiles: Definition, Formulas, Inter Quartile Range, Examples (2)
(Video) Quartiles & Interquartile Range

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: Definition, Formulas, Inter Quartile Range, Examples (3)

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.

Quartiles: Definition, Formulas, Inter Quartile Range, Examples (4)
  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:

(Video) Quartiles - Definition, Formulas, Interquartile Range | What is Quartile | ICSE Class 10 Math

  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 \(=7\).
Quartiles: Definition, Formulas, Inter Quartile Range, Examples (5)

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:
Quartiles: Definition, Formulas, Inter Quartile Range, Examples (6)

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.

Quartiles: Definition, Formulas, Inter Quartile Range, Examples (7)

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).

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

(Video) How to Find the Interquartile Range of a Set of Data | Statistics

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.

Quartiles: Definition, Formulas, Inter Quartile Range, Examples (8)

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)

Quartiles: Definition, Formulas, Inter Quartile Range, Examples (9)

Learn Concepts on Statistics

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 \(=7\).

Quartiles: Definition, Formulas, Inter Quartile Range, Examples (10)

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 \(=Q_{3}-Q_{1}=77-64=13\).

Quartiles: Definition, Formulas, Inter Quartile Range, Examples (11)

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.

(Video) How To Find The Interquartile Range & any Outliers - Descriptive Statistics

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 \(=\frac{\left[\left(\frac{n}{2}\right)^{\text {th}} \text { term }+\left(\frac{n+1}{2}\right)^{\text {th}} \text { term }\right]}{2}\).

Now you are provided with all the necessary information on the concept of quartiles and we hope this detailed article is helpful to you. If you have any queries regarding this article, please ping us through the comment section below and we will get back to you as soon as possible.

(Video) Finding Q1,Q3, and IQR ( Interquartile Range)

FAQs

What is interquartile range with example? ›

There are 5 values below the median (lower half), the middle value is 64 which is the first quartile. There are 5 values above the median (upper half), the middle value is 77 which is the third quartile. The interquartile range is 77 – 64 = 13; the interquartile range is the range of the middle 50% of the data.

What is Q1 and Q3 in interquartile range? ›

The lower quartile, or first quartile (Q1), is the value under which 25% of data points are found when they are arranged in increasing order. The upper quartile, or third quartile (Q3), is the value under which 75% of data points are found when arranged in increasing order.

What is Q1 interquartile range? ›

Calculate the interquartile range

Q1 is the value below which 25 percent of the distribution lies, while Q3 is the value below which 75 percent of the distribution lies. You can think of Q1 as the median of the first half and Q3 as the median of the second half of the distribution.

What is the formula for quartiles? ›

First Quartile(Q1)=((n+1)/4)th Term also known as the lower quartile. The second quartile or the 50th percentile or the Median is given as: Second Quartile(Q2)=((n+1)/2)th Term. The third Quartile of the 75th Percentile (Q3) is given as: Third Quartile(Q3)=(3(n+1)/4)th Term also known as the upper quartile.

What are the 4 quartiles? ›

Quartiles are the values that divide a list of numerical data into three quarters.
...
Quartiles in Statistics
  • First quartile: 25% from smallest to largest of numbers.
  • Second quartile: between 25.1% and 50% (till median)
  • Third quartile: 51% to 75% (above the median)
  • Fourth quartile: 25% of largest numbers.

What is quartiles in math? ›

Numbers separating an ordered distribution into four groups, each containing (as closely as possible) equal numbers of values. The most common names for these three numbers, in order from lowest to highest, are lower quartile, median and upper quartile.

How do you explain interquartile range? ›

The interquartile range (IQR) is the distance between the first quartile (Q1) and the third quartile (Q3). 50% of the data are within this range. For this ordered data, the interquartile range is 8 (17.5–9.5 = 8). That is, the middle 50% of the data is between 9.5 and 17.5.

How do you write median and interquartile range? ›

Answer: Median is generally represented in APA style as Mdn; numerically, median and interquartile range are represented as "median (IQR)" – that is, with the median outside the parentheses and the interquartile range within.

What is Q1 q2 and Q3 in math? ›

Now, we can map out the four groups formed from the quartiles. The first group of values contains the smallest number up to Q1; the second group includes Q1 to the median; the third set is the median to Q3; the fourth category comprises Q3 to the highest data point of the entire set.

What is the interquartile range calculator? ›

This simple tool works out the interquartile range of a set of numbers by calculating the 25th and 75th percentiles, and then subtracting the former from the latter (i.e., IQR = Q3 - Q1). Enter your data into the text box below, and then hit the "Calculate Percentile" button.

What does interquartile range in math? ›

more ... The range from Quartile 1 to Quartile 3: Q3 − Q1. (Quartiles are the values that divide a list of numbers into quarters.)

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