Quartiles

Quartiles divide a data set into four equal parts, each containing 25% of the data. They help to understand the spread and center of the data. As an important concept in statistics, quartiles are used to analyze large data sets by highlighting values near the middle. This method is particularly useful for identifying outliers and comparing different data sets.

Quartiles are values that split lists of datasets into quarters, resulting in lower, middle, and upper segments.

Lower or First Quartile (Q₁)

• Quartile 1 lies between the starting term and the middle term.
• This is the median of the lower half of the data set.
• It is also known as the 25th percentile because it marks the point where 25% of the data is below it.

Median or Second Quartile (Q₂)

• Quartile 2 lies between the starting terms and the last terms, i.e., the Middle term.
• This is the median of the entire data set.
• It is also known as the 50th percentile, as it divides the data into two halves.

Upper or Third Quartile

• Quartile 3 lies between quartile 2 and the last term.
• This is the median of the upper half of the data set.
• It is also known as the 75th percentile because it marks the point where 75% of the data is below it.

Quartile Formula

As mentioned above, Quartile divides the data into 4 equal parts. There is a separate formula for finding each quartile value, and the steps to obtain the quartile formula are as shown below as follows: 

Step 1: Sort the given data in ascending order.

Step 2: Find respective quartile values/terms as per need from the below formulae.

• First Quartile = \frac{(n + 1)}{4} \text{\small th term}
• Second Quartile = \frac{(n + 1)}{2} \text{\small th term}
• Third Quartile = \frac{3(n + 1)}{4} \text{\small th term}

Where n is the total number of values in the dataset.

Example: Find the Q₁, Q₂, and Q₃ of the given dataset: 3, 5, 7, 8, 10, 11, 3, 1, 1, 11.

Solution:

Arrange the dataset in ascending or descending order, depending on your preference. We will arrange the data in ascending order: 1, 3, 3, 5, 7, 8, 10, 11, 11

Cut the list into Quarters: (n = number of terms)

• Quartile 1 (Q1) = [(n + 1)/4] ^th Term = [( 9+ 1) / 4] = 2.25 term [Rounds off to 3 term] = 3
• Quartile 2 (Q2) = [(n + 1)/2 ] ^th Term = [{9 + 1)/2] = 5 ^th Term = 7
• Quartile 3 (Q3) = [3(n + 1)/4 ] ^th Term = [3 (10 + 1)/4] ^th Term = 7.5 ^th Term [Rounds off to 8 ^th Term] = 11

In the chart, the continuous variable is plotted against the independent variable, and the data is divided into quartiles.

Generalized Formula for Quartile

The generalized formula for the quartile is,

\bold{\text{Quartile}_r = l_1 + ( i\cdot \frac{n}{4} - c_f) \cdot \frac{(l_2-l_1)}{f}}

Where,

• Quartile_r indicates r^th quartile,
• l₁, l₂ are the lower and upper limit value that contains the i^th quartile,
• f is the frequency count,
• c_f is the cumulative frequency of the class preceding the quartile class.

Using this generalized formula, the first and third quartiles can be calculated as:

• \bold{\text{Q}_1 = l_1 + ( \frac{n}{4} - c_f) \cdot \frac{(l_2-l_1)}{f}}
• \bold{\text{Q}_3 = l_1 + ( \frac{3n}{4} - c_f) \cdot \frac{(l_2-l_1)}{f}}

Interquartile Range

Interquartile Range is the distance between the first quartile and the third quartile. It is also known as a mid-spread. It helps us to calculate variation for the data, which is divided into quartiles. The formula for calculating the Interquartile range is given by,

Interquartile Range (IQR) = Q₃ - Q₁

Where, 

• Q₃ is the third/upper quartile, and 
• Q₁ is the first/lower quartile.

IQR is used for:

1. Identify Outliers: Since the IQR focuses on the middle 50% of the data, any values that fall below Q1 - 1.5*IQR or above Q3 + 1.5*IQR are typically considered outliers.
2. Measure Variability: It helps in understanding the spread of data around the median, giving us a better sense of data distribution than the range which can be influenced by outliers.
3. Statistical Analysis: The IQR is often used in boxplots to visualize the spread and detect outliers. It is also useful for comparing different datasets, especially when the data contains outliers.

Note: Outliers are data points that significantly differ from the majority of the data, often appearing as extreme values far from the rest.

Quartile Deviation 

Quartile Deviation is defined as half of the distance between the first quartile and the third quartile. It is also known as the semi-interquartile range. The formula for quartile deviation is given by,

Quartile Deviation = (Q₃ - Q₁)/2

Quartile vs Percentile

The key differences between Quartile and Percentile are given as follows:

Related Articles

• Central Tendency
• Mean
• Mode

Solved Problems

Problem 1: Find Quartile 1 for the given data: 10, 30, 5, 12, 20, 40, 25, 15, 18.

Solution:

Step 1: Sort the given data in ny order ( ascending order / descending order) 
5, 10, 12, 15, 18, 20, 25, 30, 40

Step 2: Find 1st Quartile
FIrst Quartile = (\frac{n + 1}{4})^{th}           term

Here n = 9 because there are total 9 numbers in the given data.
⇒ First Quartile = ((9 + 1)/4)^th term
⇒ First Quartile = (10/4)^th term
⇒ First Quartile = 2.5^th term

Now, 2.5^th term = 2nd term + (0.5) (3^rd term - 2^nd term)
⇒ 2.5^th term = (10) + (0.5) (12 - 10)
⇒ 2.5^th term = 10+1 
⇒ 2.5^th term = 11

The First Quartile value is 11.

Problem 2: Find the Second Quartile for the data 10, 30, 5, 12, 20, 40, 25, 15, 18.

Solution:

Step 1: Sort the given data in the ascending order

5, 10, 12, 15, 18, 20, 25, 30, 40

Step 2: Find 2nd Quartile

Second Quartile = (\frac{n + 1}{2})^{th}           term

Here n = 9 because there are total 9 numbers in the given data.
⇒ Second Quartile = (\frac{9 + 1}{2})^{th}           term
⇒ Second Quartile = (10/2)^th term
⇒ Second Quartile = 5^th term
5^th term is 18

So the Second Quartile value is 18.

Problem 3: Find the third Quartile for the data 10, 30, 5, 12, 20, 40, 25, 15, 18.

Solution:

Step 1: Sort the given data in the ascending order
5, 10, 12, 15, 18, 20, 25, 30, 40

Step 2: Find 3^rd Quartile
Third Quartile = \frac{3(n + 1)}{4}^{th}    term

Here n = 9 because there are total 9 numbers in the given data.
⇒ Third Quartile = \frac{3(n + 1)}{4}^{th}     term
⇒ Third Quartile= \frac{3 \times (10)}{4}^{th}     term
⇒ Third Quartile= 7.5^th term

7.5^th term is average result of 7^th and 8^th term = (25 + 30)/2 = 27.5
Remember:  7.5^th term = 7^th term + (0.5) (8^th term - 7^th term)

The most recommended method to find value is mentioned above
Because the term not always N.5 something  it may vary from N.1 to N.9 
Here, N be any natural number.

So the third Quartile value is 27.5.

Problem 4: Find the first, second, and third quartiles for the data 8, 5,15,  20, 18, 30,  40, 25

Solution:

Step 1: Sort the given data in ascending order

5, 8, 15, 18, 20, 25, 30, 40

Step 2: Find all Quartiles step by step

First Quartile = {(n + 1)/4}th term

Here n = 8 because there are total 8 numbers in the given data.

⇒ First Quartile = {(8 + 1)/4}th term
⇒ First Quartile = (9/4)th term
⇒ First Quartile = 2.25th term

Thus, 2.25th term = 2nd term + (0.25)(3rd term − 2nd term)

⇒ 2.25th term = 8 + (0.25)(15 − 8)
⇒ 2.25th term = 8 + (0.25 × 7) = 8 + 1.75 = 9.75

First Quartile value is 9.75

Second Quartile = {(n + 1)/2}th term

⇒ Second Quartile = {(8 + 1)/2}th term
⇒ Second Quartile = (9/2)th term
⇒ Second Quartile = 4.5th term

Thus, 4.5th term = 4th term + (0.5)(5th term − 4th term)

⇒ 4.5th term = 18 + (0.5)(20 − 18)
⇒ 4.5th term = 18 + (0.5 × 2) = 18 + 1 = 19

So the Second Quartile value is 19

Third Quartile = {3(n + 1)/4}th term

⇒ Third Quartile = {3(8 + 1)/4}th term
⇒ Third Quartile = (27/4)th term
⇒ Third Quartile = 6.75th term

Thus, 6.75th term = 6th term + (0.75)(7th term − 6th term)

⇒ 6.75th term = 25 + (0.75)(30 − 25)
⇒ 6.75th term = 25 + (0.75 × 5) = 25 + 3.75 = 28.75

So the Third Quartile value is 28.75

Problem 5: What is the Interquartile Range for the data if the first quartile is 10 and the third quartile is 30cm?

Solution:

Given,

• Q₁ = 10
• Q₃ = 30

Interquartile range = Q₃ - Q₁
⇒ Interquartile range = 30 - 10

Thus, Interquartile range is 20.

Problem 6: What is the Quartile Deviation for the data if the first quartile is 15 and the third quartile is 30cm?

Solution:

Given,

• Q₁ = 15
• Q₃ = 30

Quartile Deviation = (Q₃ - Q₁)/2

⇒ Quartile Deviation = (30 - 15)/2
⇒ Quartile Deviation = 15/2

Thus, Quartile Deviation is 7.5

Practice Questions

Question 1: Given the dataset: 10, 30, 5, 12, 20, 40, 25, 15, 18, calculate Q1, Q2, and Q3.

Question 2: Annual salaries (in thousands) of 20 employees: 22, 25, 28, 29, 32, 34, 36, 37, 39, 41, 43, 45, 47, 50, 52, 54, 56, 59, 61, 63. Determine Q1, Q2, and Q3.

Question 3: Given the dataset: 3, 8, 12, 17, 20, 24, 27, 31, compute Q1, Q2, and Q3.

Question 4: Data points: 1, 2, 2, 3, 4, 4, 5, 5, 6, 7, 100. Identify Q1, Q2, and Q3 and discuss how the outlier affects the quartile values.

Question 5: Test scores of 15 students: 55, 60, 61, 63, 67, 69, 72, 75, 78, 81, 83, 85, 88, 90, 92. Calculate Q1, Q2, and Q3 and interpret the results to understand the spread and distribution of the students' test scores.