Quartile Deviation Formula

Quartile Deviation Formula (Table of Contents)

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What is Quartile Deviation Formula?

 The Quartile Deviation(QD) is the product of half of the difference between the upper and lower quartiles. Mathematically we can define 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 Quartile Deviation Formula in a better manner.

Quartile Deviation Formula – Example #1

The number of complaints lodged against the steal of the 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, the way that it lies halfway between the lowest value and the median; where the third quartile lies halfway between the median and the largest value.

First Quartile (Q₁) is calculated using the formula given 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) is calculated as

• 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 

Third Quartile (Q₃) is calculated using the formula given below

Third Quartile (Q₃)

Q_i= [i * (n + 1) /4] ^th obsevation

• Q₃ = [1 * (n + 1) /4] ^th obsevation
• Q₃ = [(10 + 1) /4] ^th obsevation
• 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) is calculated as

• Q₃ = 8^th obsevation + 0.25 * (9^th obsevation – 8^th obsevation)
• 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 –

Quartile Deviation is calculated using the formula given below

Quartile Deviation = (Q₃ – Q₁) / 2

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

Coefficient of Quartile Deviation is calculated using the formula given 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 shows the one-day sales of a shopping mall, where we determine the frequency of the first 50 customers of different age group. 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

First Quartile (Q₁) is calculated using the formula given below

First Quartile (Q₁) 

Q_i = [ i * (N) /4 ]^th obsevation

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

Since 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

Third Quartile (Q₃) is calculated using the formula given below

Third Quartile (Q₃)

Q_i = [ i * (N) /4 ]^th obsevation

Q1 =[ i* (N) /4 ]^th obsevation

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

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

Therefor 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:

Quartile Deviation is calculated using the formula given below

Quartile Deviation = (Q₃ – Q₁) / 2

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

Coefficient of Quartile Deviation is calculated using the formula given 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. As 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 which is also known as Quartile deviation. Now, we can calculate quartile deviation for both grouped and ungrouped data by using a formula given 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 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 obsevation

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

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 calculate the third Quartile is given as:

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

• Q₃ = i * (n+1) /4

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

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

15^th observation = 90

16^th obsevation = 96

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

Step 3: Calculate the Quartile Deviation & Coefficient of Quartile Deviation on the basis of 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 take into consideration 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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