Quartile Deviation Formula

Updated July 26, 2023

Quartile Deviation Formula (Table of Contents)

• Formula
• Examples
• Calculator

What is Quartile Deviation Formula?

 The Quartile Deviation(QD) is the product of half the difference between the upper and lower quartiles.

Mathematically we can define it as:

A Coefficient of QD is used to study & compare the degree of variation in different situations.

Examples of Quartile Deviation Formula (With Excel Template)

Let’s take an example to understand the calculation of the Quartile Deviation Formula in a better manner.

Quartile Deviation Formula – Example #1

The number of complaints lodged against the stealing of vehicles in a day was calculated for the next 10 days. And the data is given below. Calculate the Quartile Deviation and its coefficient for the given discrete distribution case.

Solution:

Arrange the data in Ascending Order

Now, we will find the first quartile, which lies halfway between the lowest value and the median, where the third quartile lies halfway between the median and the largest value.

The formula to calculate First Quartile (Q₁) is as below:

First Quartile (Q₁) 

Q_i= [i * (n + 1) /4] ^th observation

Q₁= [1 * (10 + 1) /4] ^th observation

Q₁ = [1 * (10 + 1) /4] ^th observation

Q₁ = 2.75^th observation

So, 2..75^th observation lies between the 2^nd and 3^rd value in the ordered group, or midways between 12 & 14 therefore

First Quartile (Q₁) calculation:

• Q₁ = 2^nd observation + 0.75 * (3^rd observation – 2^nd observation)
• Q₁ = 12 + 0.75 * (14 – 12)
• Q₁ = 12 + 1.50
• Q₁ = 13.50 

The formula to calculate Third Quartile (Q₃) is as below:

Third Quartile (Q₃)

Q_i= [i * (n + 1) /4] ^th observation

• Q₃ = [1 * (n + 1) /4] ^th observation
• Q₃ = [(10 + 1) /4] ^th observation
• Q₃ = 8.25^th observation

So, 8..25^th observation lies between the 8^th and 9^th value in the ordered group, or midways between 30 & 35 therefore

Third Quartile (Q₃) calculation:

• Q₃ = 8^th observation + 0.25 * (9^th observation – 8^th observation)
• Q₃ = 30 + 0.25 * (35 – 30)
• Q₃ = 31.25

Now using the Quartile values Q1 & Q3. We will calculate its Quartile deviation & its coefficient as follows –

The formula to calculate Quartile Deviation is as below:

Quartile Deviation = (Q₃ – Q₁) / 2

• Quartile Deviation =(31.25 – 13.50) / 2
• Quartile Deviation = 8.875

The formula to calculate the Coefficient of Quartile Deviation is as below:

Coefficient of Quartile Deviation = (Q₃ – Q₁) / (Q₃ + Q₁)

• Coefficient of Quartile Deviation = (31.25 – 13.50) /(31.25 + 13.50)
• Coefficient of Quartile Deviation =0. 397

Quartile Deviation Formula – Example #2

Following are the observations showing the one-day sales of a shopping mall, where we determine the frequency of the first 50 customers of different age groups. Now, we need to Calculate the quartile deviation and coefficient of quartile deviation.

Solution:

In the case of Frequency Distribution, Quartiles can be calculated by using the formula:

Q_i = l + (h / f) * ( i * (N/4) – c) ; i = 1,2,3

Where,

• l = Lower Boundary of Quartile Group
• h =Width of Quartile Group
• f = Frequency of Quartile Group
• N = Total Number of Observations
• c = Cumulative Frequency

First, we have to calculate the cumulative frequency table

The formula to calculate First Quartile (Q₁) is as below:

First Quartile (Q₁) 

Q_i = [ i * (N) /4 ]^th observation

• Q₁ = [1 * (50) / 4]^th observation
• Q₁ = 12.50 ^th observation

Since the 12.50^th value is in the interval 44.5 – 49.5

Therefore Group of Q1 is (44.5 – 49.5)

Q_i = l + (h / f) *( i * (N/4) – c)

• Q₁ = (44.5 + ( 5 /8) * (1 * (50 / 4) – 5)
• Q₁ = 44.5 + 4.6875
• Q₁ = 49.19

The formula to calculate Third Quartile (Q₃) is as below:

Third Quartile (Q₃)

Q_i = [ i * (N) /4 ]^th observation

Q1 =[ i* (N) /4 ]^th observation

• Q₃= [3 * (50) / 4]^th observation
• Q₃ = 37.50 ^th observation

Since 37.50^th value is in the interval (59.5 – 64.5)

Therefore Group of Q3 is (59.5 – 64.5)

Q_i = l + (h / f) *( i * (N/4) – c)

• Q₃ = 59.5 + (5 /9) * (3 * (50/4 ) – 34)
• Q₃ = 59.5 + 1.944
• Q₃ = 61.44

By putting the values into the formulas of quartile deviation and coefficient of quartile deviation, we get:

The formula to calculate Quartile Deviation is as below:

Quartile Deviation = (Q₃ – Q₁) / 2

•  Quartile Deviation = (61.44 – 49.19) /2
• Quartile Deviation = 6.13

The formula to calculate the Coefficient of Quartile Deviation is as below:

Coefficient of Quartile Deviation = (Q₃ – Q₁) / (Q₃ + Q₁)

• Coefficient of Quartile Deviation = (61.44 – 49.19) / (61.44 + 49.19)
• Coefficient of Quartile Deviation = 12.25 / 110.63
• Coefficient of Quartile Deviation = 0.11

Explanation

Quartile deviation is the dispersion in the middle of the data where. It defines the spread of the data. We know that the difference between the Third Quartiles and First Quartiles is called the Interquartile range, and half of the Interquartile Range is called Semi-Interquartile, also known as the Quartile deviation. Now, we can calculate the quartile deviation for both grouped and ungrouped data using the formula below.

Quartile Deviation = (Third Quartile – First Quartile) / 2

Quartile Deviation =(Q₃ – Q₁) / 2

While the coefficient of quartile deviation is used to compare the variation between two data sets.6687 Moreover, quartile deviation is not affected by the extreme values where it contains extreme values. A Coefficient of Quartile deviation can be calculated in such a fashion.

Coefficient of Quartile Deviation = (Q₃ – Q₁) / (Q₃ + Q₁)

The concept of quartile deviation and coefficient of quartile can be explained with the help of an example in a definite set of steps.

Step 1: Obtain a set of Ungrouped data

In the problem statement, we have considered runs scored by a batsman in the last 20 test matches: 96, 70,100, 89,78,56,45,78,68,42,66,89,90,54,44,67,87,90,97, and 98

Step 2: Arrange the data in ascending order:

42,44,45,54,56,66,67,68,70,78,78,87,89,89,90,92,96,97,98,100

First Quartile(Q₁)

Calculate the first quartile

Q_i= i * (n+1) /4^th observation

• Q₁ = 1 * (20 + 1) /4 ^th observation
• Q₁ = 5.25^th observation

So, 5.25^th observation lies between the 5^th and 6^th value in the ordered group, or midways between 55 & 66 therefore

• Q₁= 55 + 0.25 * (66 – 55)
• Q₁ = 55 + 2.75
• Q₁= 57.25

Third Quartile (Q₃)

The calculation of the third Quartile is given as follows:

Q_i= i * (n+1) /4th observation

• Q₃ = i * (n+1) /4

• Q₃ = 3 * (20 + 1) /4^th observation
• Q₃ = 15.75^th observation

Where 15.75^th lies between the 15^th and 16^th value in the ordered group

15^th observation = 90

16^th observation = 96

• Q₃ = 90 +0.75 * (96 – 90)
• Q₃ = 90 + 4.5
• Q₃ = 94.5

Step 3: Calculate the Quartile Deviation & Coefficient of Quartile Deviation based on the respective outcome.

Quartile Deviation =(Q₃ – Q₁) / 2

• Quartile Deviation = (94.5 – 57.25) / 2
• Quartile Deviation = 18.625

Coefficient of Quartile Deviation = (Q₃ – Q₁) / (Q₃ + Q₁)

• Coefficient of Quartile Deviation = (94.5 – 57.25) / (94.5 +57.25)
• Coefficient of Quartile Deviation = 0.2454

Relevance and Uses of Quartile Deviation Formula

• The Quartile Deviation doesn’t consider much more extreme points of the distribution.
• QD also changes with respect to the change of scale of data.
• It is the best measure for the open-ended system.
• Less affected by the sampling fluctuations in the dataset
• Solely depend on the central values in the distribution.

Quartile Deviation Formula Calculator

You can use the following Quartile Deviation Formula Calculator

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