Quartile Practice Problems

Quartiles are values that divide an ordered dataset into four equal parts, with each part containing approximately 25% of the observations.

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