Quartile Deviation Formula (Table of Contents)
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:
Quartile Deviation defines the absolute measure of dispersion. Whereas the relative measure corresponding to QD, is known as the coefficient of QD, which is obtained by applying the certain set of the formula:
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.
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First Quartile (Q_{1}) is calculated using the formula given below
First Quartile (Q_{1})
Q_{i}= [i * (n + 1) /4] ^{th} observation
Q_{1}= [1 * (10 + 1) /4] ^{th} observation
Q_{1 }= [1 * (10 + 1) /4] ^{th} observation
Q_{1 }= 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 (Q_{1}) is calculated as
 Q_{1 }= 2^{nd} observation + 0.75 * (3^{rd} observation – 2^{nd} observation)
 Q_{1 }= 12 + 0.75 * (14 – 12)
 Q_{1 }= 12 + 1.50
 Q_{1 }= 13.50
Third Quartile (Q_{3}) is calculated using the formula given below
Third Quartile (Q_{3})
Q_{i}= [i * (n + 1) /4] ^{th} obsevation
 Q_{3 }= [1 * (n + 1) /4] ^{th} obsevation
 Q_{3 }= [(10 + 1) /4] ^{th} obsevation
 Q_{3 }= 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 (Q_{3}) is calculated as
 Q_{3 }= 8^{th} obsevation + 0.25 * (9^{th} obsevation – 8^{th} obsevation)
 Q_{3 }= 30 + 0.25 * (35 – 30)
 Q_{3 }= 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 = (Q_{3} – Q_{1}) / 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 = (Q_{3} – Q_{1}) / (Q_{3} + Q_{1})
 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 oneday 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 (Q_{1}) is calculated using the formula given below
First Quartile (Q_{1})
Q_{i }= [ i * (N) /4 ]^{th} obsevation
 Q_{1 }= [1 * (50) / 4]^{th} obsevation
 Q_{1 }= 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)
 Q_{1 }= (44.5 + ( 5 /8) * (1 * (50 / 4) – 5)
 Q_{1 }= 44.5 + 4.6875
 Q_{1 }= 49.19
Third Quartile (Q_{3}) is calculated using the formula given below
Third Quartile (Q_{3})
Q_{i }= [ i * (N) /4 ]^{th} obsevation
Q1 =[ i* (N) /4 ]^{th} obsevation
 Q_{3}= [3 * (50) / 4]^{th} obsevation
 Q_{3 }= 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)
 Q_{3 }= 59.5 + (5 /9) * (3 * (50/4 ) – 34)
 Q_{3 }= 59.5 + 1.944
 Q_{3 }= 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 = (Q_{3} – Q_{1}) / 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 = (Q_{3} – Q_{1}) / (Q_{3} + Q_{1})
 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 SemiInterquartile 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 =(Q_{3} – Q_{1}) / 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 = (Q_{3} – Q_{1}) / (Q_{3} + Q_{1})
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(Q_{1})
Calculate the first quartile
Q_{i}= i * (n+1) /4^{th} obsevation
 Q_{1 }= 1 * (20 + 1) /4 ^{th} obsevation
 Q_{1 }= 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
 Q_{1}= 55 + 0.25 * (66 – 55)
 Q_{1} = 55 + 2.75
 Q_{1}= 57.25
Third Quartile (Q_{3})
The calculate the third Quartile is given as:
Q_{i}= i * (n+1) /4 th obsevation
 Q_{3 }= i * (n+1) /4
 Q_{3 }= 3 * (20 + 1) /4^{th} observation
 Q_{3 }= 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
 Q_{3 }= 90 +0.75 * (96 – 90)
 Q_{3 }= 90 + 4.5
 Q_{3 }= 94.5
Step 3: Calculate the Quartile Deviation & Coefficient of Quartile Deviation on the basis of the respective outcome.
Quartile Deviation =(Q_{3} – Q_{1}) / 2
 Quartile Deviation = (94.5 – 57.25) / 2
 Quartile Deviation = 18.625
Coefficient of Quartile Deviation = (Q_{3} – Q_{1}) / (Q_{3} + Q_{1})
 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 openended 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
Q_{3}  
Q_{1}  
Quartile Deviation  
Quartile Deviation = 


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