(I) Quartiles

There are several ways to divide an observation when required. To divide the observation into two equally-sized parts, the median can be used. A quartile is a set of values that divides a dataset into four equal parts. The first quartile, second quartile, and third quartile are the three basic quartile categories. The lower quartile is another name for the first quartile and is denoted by the letter Q₁. The median is another term for the second quartile and is denoted by the letter Q₂. The third quartile is often referred to as the upper quartile and is denoted by the letter Q₃.

Simply put, the values which divide a list of numerical data into three quarters are known as quartiles. The center of the three quadrants measures the central point of distribution and displays data that are close to the center. The quartiles represent the distribution or dispersion of the data set as a whole. The formulas for calculating quartiles are:

[Tex]Q_{1}=[\frac{N+1}{4}]^{th}~item [/Tex]

[Tex]Q_{2}=[\frac{N+1}{2}]^{th}~item [/Tex]

[Tex]Q_{3}=[\frac{3(N+1)}{4}]^{th}~item [/Tex]

where, n is the total number of observations, Q₁ is First Quartile, Q₂ is Second Quartile, and Q₃ is Third Quartile.

Example 1: 

Calculate the lower and upper quartiles of the following weights in the family: 25, 17, 32, 11, 40, 35, 13, 5, and 46.

Solution: 

First of all, organise the numbers in ascending order.

5, 11, 13, 17, 25, 32, 35, 40, 46

Lower quartile, [Tex]Q_{1}=[\frac{N+1}{4}]^{th}~item [/Tex]

[Tex]Q_{1}=[\frac{9+1}{4}]^{th}~item [/Tex]

Q₁ = 2.5^th term

As per the quartile formula;

Q₁ = 2^nd term + 0.5(3^rd term – 2^nd term)

Q₁ = 11 + 0.5(13 – 11) = 12

Q₁ = 12

Upper Quartile, [Tex]Q_{3}=[\frac{3(N+1)}{4}]^{th}~item [/Tex]

[Tex]Q_{3}=[\frac{3(9+1)}{4}]^{th}~item [/Tex]

Q₃ = 7.5^th item

Q₃ = 7^th term + 0.5(8^th term – 7^th term)

Q₃ = 35 + 0.5(40 – 35) = 37.5

Q₃ = 37.5

Example 2:

Calculate Q1 and Q3 for the data related to the age in years of 99 members in a housing society.

Solution:

[Tex]Q_{1}=[\frac{N+1}{4}]^{th}~item [/Tex]

[Tex]Q_{1}=[\frac{99+1}{4}]^{th}~item [/Tex]

Q₁ = 25^th item

Now, the 25^th item falls under the cumulative frequency of 25 and the age against this cf value is 18.

Q₁ = 18 years

[Tex]Q_{3}=[\frac{3(N+1)}{4}]^{th}~item [/Tex]

[Tex]Q_{3}=[\frac{3(99+1)}{4}]^{th}~item [/Tex]

Q₃ = 75^th item

Now, the 75^th item falls under the cumulative frequency of 85 and the age against this cf value is 40.

Q₃ = 40 years

Example 3: 

Determine the quartiles Q₁ and Q₃ for the company’s salary listed below.

Solution:

[Tex]Q_{1}~Class=\frac{N}{4} [/Tex]

[Tex]Q_{1}~Class=\frac{60}{4} [/Tex]

= 15^th item

Now, the 15^th item falls under the cumulative frequency 22 and the salary against this cf value lies in the group 600-700.

[Tex]Q_{1}=l_{1}+\frac{\frac{N}{4}-m_{1}}{f_{1}}\times{c_{1}} [/Tex]

[Tex]Q_{1}=600+\frac{\frac{60}{4}-10}{12}\times{100} [/Tex]

Q₁ = ₹641.67

[Tex]Q_{3}~Class=\frac{3N}{4} [/Tex]

[Tex]Q_{3}~Class=\frac{180}{4} [/Tex]

Q₃ = 45^th item

Now, the 45^th item falls under the cumulative frequency 52 and the salary against this cf value lies in the group 800-900.

[Tex]Q_{3}=l_{1}+\frac{\frac{3N}{4}-m_{3}}{f_{3}}\times{c_{3}} [/Tex]

[Tex]Q_{3}=800+\frac{\frac{180}{4}-38}{14}\times{100} [/Tex]

[Tex]Q_{3}=800+\frac{7}{14}\times{100} [/Tex]

[Tex]Q_{3}=800+50 [/Tex]

Q₃ = ₹850