Question

Q1=30, Q3=50. What is the value of co-efficient of quartile deviation?​

Answer 1

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• The coefficient of quartile deviation (sometimes called the quartile coefficient of dispersion) allows you to compare dispersion for two or more sets of data. The formula is: If one set of data has a larger coefficient of quartile deviation than another set, then that data set's interquartile dispersion is greater.

Answer 2

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 = (Q3 – Q1) / 2

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:

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

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

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

First Quartile (Q1)

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

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

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

Q1 = 2.75th observation

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

between 12 & 14 therefore

First Quartile (Q1) is calculated as

Calculation of First Quartile -1.3

Q1 = 2nd observation + 0.75 * (3rd observation – 2nd 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)

Qi= [i * (n + 1) /4] th obsevation

Q3 = [1 * (n + 1) /4] th obsevation

Q3 = [(10 + 1) /4] th obsevationQ3 = 8.25th observation

So, 8..25th observation lies between the 8th and 9th value in the ordered group, or midways

between 30 & 35 therefore

Third Quartile (Q3) is calculated as

Calculation of Third Quartile-1.4

Q3 = 8th obsevation + 0.25 * (9th obsevation – 8th 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 Formula-1.5

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 Q.D Formula-2.6

Coefficient of Quartile Deviation = (31.25 – 13.50) /(31.25 + 13.50)

Coefficient of Quartile Deviation =0. 397

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.

Quartile Deviation Formula-2.1

Solution:

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

Qi = 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 GroupN = Total Number of Observations

c = Cumulative Frequency

First, we have to calculate the cumulative frequency table

cumulative frequency table -2.2

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

First Quartile (Q1)

Qi = [ i * (N) /4 ]th obsevation

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

Q1 = 12.50 th obsevation

Since 12.50th value is in the interval 44.5 – 49.5

Therefore Group of Q1 is (44.5 – 49.5)

Qi = l + (h / f) *( i * (N/4) – c)

Q.D Formula-2.3

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)

Qi = [ i * (N) /4 ]th obsevation

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

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

Q3 = 37.50 th obsevation

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

Therefor Group of Q3 is (59.5 – 64.5)

Qi = l + (h / f) *( i * (N/4) – c)

Quartile Deviation Formula-2.4

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 belowQuartile Deviation = (Q3 – Q1) / 2

Quartile Deviation Formula-2.5

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)

Quartile Deviation Formula-2.6

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)