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# Quartiles

## Introduction to Quartiles

Last updated date: 21st Mar 2023
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Quartiles are a set of values with three points that divide a data set into four identical portions in statistics. In statistics, we usually deal with a lot of numerical data. There are a number of concepts and formulas that can be used in a variety of studies and surveys. The values that divide a list of numerical data into three quarters are called quartiles. The middle quarter of the three quarters measures the distribution's central point and displays statistics that are close to it. The lower half of the quarters represent the information set that falls below the median, while the upper half represents the information set that falls above the median. The quartiles represent the data set's distribution or dispersion in general. In this article, we are going to discuss Quartile, its meaning, formula, definition, how to calculate it, solved examples and also a few frequently asked questions will also be answered.

## What is a Quartile?

Quartiles in statistics are the values, which divide the number of data points into four parts, or quarters, of more or less equal size. To calculate quartiles, the data must be arranged in ascending order i.e. from smallest to largest. Following are the three main quartiles.

• The first quartile, or $Q_1$ often known as the lower quartile, divides the data into two groups: the lowest 25% and the highest 75%.

• The second quartile, or $Q_2$ often known as the middle quartile or the median, divides data into two equal portions. In the second quartile, 50% of the data lies below this point.

• The third quartile, or $Q_3$ often known as the upper quartile, divides the top 25% of data from the bottom 75%. The third quartile is also known as the upper or 75th empirical quartile as 75% of the data lies below this point.

## Quartile Meaning

Quartiles are the values that split the list of numbers into quarters.

• Arrange the list of numbers in order.

• Then cut the list of numbers into four equal parts.

• The quartiles are found at the “cuts”.

Example: 2, 4, 5, 7, 6, 9, 3

Arrange the numbers in order: 2, 3, 4, 5, 6, 7, and 9.

Cut the given list of numbers into quarters as shown below:

Here,

The result is:

• Quartile 1 $(Q_1)$or Lower Quartile = 3

• Quartile 2 $(Q_2)$ or Middle Quartile = 5, which is also the median.

• Quartile 3 $(Q_3)$ or Upper Quartile = 7

## Quartile Formula

The quartile formula helps to divide the set of observations into four equal parts. The first quartile lies between the smallest number and median of the data set. The second quartile is the median of the data set. The third quartile lies between the highest value and median of the given data set.

When the given data set is arranged in ascending order, the quartile formulas are given as:

• $Q_1$ or Lower Quartile Formula - $\frac{(n+1)}{4}^{th}$

• $Q_2$ or MedianFormula - $\frac{(n+1)}{2}^{th} term - \frac{(n+1)}{2}^{th}$

• $Q_3$ or Upper Quartile Formula - $\frac{3(n+1)}{4}^{th}$

• The interquartile range is calculated as = Upper Quartile - Lower quartile

## How To Calculate Quartiles?

### Following are the steps to calculate quartiles in statistics:

• Step 1: Arrange the given data set in ascending order.

• Step 2: Determine how many numbers are given in the data set.

• Step 3: Set up respective quartile formulas for calculating $Q_1$ , $Q_2$ and $Q_3$

• Step 4: Substitute the value of n into the quartile formulas.

• Step 5: Complete the calculation as per the BODMAS rule.

## Solved Examples with Proven Exemplars Related to Quartiles

1. Calculate the first quartile for the data related to the final examination marks of 8 students given below:

Marks : 5, 48, 32, 52, 21, 64, 29, 57

Solution:

• Step 1: Arrange the given data in ascending order.

5, 24, 28, 32, 48, 52, 57, 64

• Step 2: Determine how many numbers are given in the data set.

The set { 5, 24, 28, 32, 48, 52, 57, 64} has 8 numbers.

• Step 3: Set up the first quartile formula for calculating Q1.

First Quartile Formula $(Q_1)$ = $(\frac{n+1}{4} )^{th}$ item

• Step 4: Substitute the value of n in the above formula

$Q_1$ = $(\frac{8+1}{4} )^{th}$ term

$Q_1$ = 2.25th term

2 term + $\frac{1}{4}$ ( 3rd term - 2nd term)

= 25 + 0.25 (28 -24)

= 25 + 1.0

= 26

Therefore , $Q_1$ = 26

2. Calculate the lower quartile of the following datasets of values: 20, 19, 21, 22, 23, 24, 25, 27, 26

Solution:

• Step 1: Arrange the given data in ascending order.

19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 31.

• Step 2: Determine how many numbers are given in the data set.

The set {19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 31} has 11 numbers.

• Step 3: Set up the lower quartile formula for calculating Q.

Lower Quartile Formula $(Q_3)$ = $\frac{3(n+1)}{4}^{th}$ item

• Step 4: Substitute the value of n in the above formula

$Q_3$ = $\frac{3(11+1)}{4}^{th}$ term

$Q_3$ = 9th term

= 27

Therefore, $Q_3$ = 27

## Multiple-Choice Questions

Q1.The upper quartile for the following set of numbers ? 26,19, 5, 7, 6, 9, 16, 12, 18, 2, 1.

1. 9th term

2. 10th term

3. 4th term

4. 8th term

Ans: The formula for upper quartile formula is $Q_3$ = $\frac{3(11+1)}{4}^{th}$ term

First put all the numbers in an ascending order: 1, 2, 5,6, 7,9,12,16,18,19,26. There area total of 11 numbers, so :

$Q_3$= $\frac{3(11+1)}{4}^{th}$ term

$Q_3$ = $\frac{3(12)}{4}^{th}$ term = 9th term.

Hence the correct option is option (A)

Q2.What is the interquartile range (IQR) of the following set of values : 20, 19 ,21, 22,23,24,25,27,26

1. IQR = 6

2. IQR = 5

3. IQR = 7

4. IQR = 8

Ans: The first step is to arrange the values in ascending order: 19, 20, 21, 22, 23, 24, 25, 26, 27.

Now put the values in the formulas above we get,

Median $(Q_2)$ = 5th term = 23

Lower quartile $(Q_1)$ = mean of the 2nd and the 3rd term =(20+21)/2 = 20.5

Upper quartile $(Q_3)$ = Mean of the 7th and 8th term =(25+26)/2 = 25.5

IQR = Upper quartile - Lower Quartile

IQR = 25.5 - 20.5 = 5

Hence, the correct answer is option (B)

## Conclusion

In statistics, quartiles are the values that divide the number of data points into four quarters or parts of more or less equal size. The quartile formula can be used to determine the value that splits a set of numbers into quarters. The information is organized in ascending order before being separated into quartiles. The median splits a set of data into two equal portions, whereas the quartiles divide it into four. We may calculate the first, second, and third quartiles using the quartile formula. Throughout the article, we have discussed thoroughly quartiles, formulas related to it and exemplar problems as well. If students want to learn more about quartile and other related topics they can visit our official website of Vedantu.

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## FAQs on Quartiles

1. What is a Quartile with an Example?

Quartile is a statistical term that defines a distribution of observations into four defined intervals based on data values, and how they compare with the whole set of observations. It is important to observe the median as a central tendency index to understand the quartile.

2. How are Quartiles Used in Statistics?

Quartiles are generally used to summarize the group of numbers. Rather than looking at a big list of numbers, quartiles give us a picture of what's going on in the list of big numbers just by observing a few numbers. Quartiles are great for describing a set of data and for making box and whisker plots. Quartiles are especially beneficial when working with data that is not symmetrically distributed or a data set that has outliers.

3. What Does Lower Quartile and Upper Quartile Refer To?

The lower quartile, also known as Q1, is the value of the middle of the first set, where 25% of the data are below this point and 75% of data are above this point.

The upper quartile, also known as Q3,  is the value of the middle of the second set, where 75% of the data are below this point and 25% are above this point.

It should be noted that the median is represented as Q2, the second quartile.