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Quartile Deviation in Continuous Series | Formula, Calculation and Examples

Last Updated : 13 Oct, 2023
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What is Quartile Deviation?

Quartile Deviation (absolute measure) divides the distribution into multiple quarters. Quartile Deviation is calculated as the average of the difference of the upper quartile (Q3) and the lower quartile (Q1).

Quartile~Deviation=\frac{Q_3-Q_1}{2}

Where,

Q3 = Upper Quartile (Size of 3[\frac{N+1}{4}]^{th}     item)

Q1 = Lower Quartile (Size of [\frac{N+1}{4}]^{th}     item)

What is Interquartile Range?

Interquartile Range refers to the difference between two quartiles.

Interquartile Range = Q3 – Q1

What is Coefficient of Quartile Deviation?

For comparative studies of the variability of two or more series with different units, the Coefficient of Quartile Deviation (relative measure) is used.

Coefficient~of~Quartile~Deviation=\frac{Q_3-Q_1}{Q_3+Q_1}

Where,

Q3 = Upper Quartile (Size of 3[\frac{N+1}{4}]^{th}        item)

Q1 = Lower Quartile (Size of [\frac{N+1}{4}]^{th}        item)

Examples of Quartile Deviation in Continuous Series

Example 1:

Calculate the interquartile range, quartile deviation, and coefficient of quartile deviation from the following figures:

Quartile Deviation in Continuous Series

 

Solution:

Quartile Deviation in Continuous Series

 

Q_1=Size~of~[\frac{N}{4}]^{th}~item=Size~of~[\frac{200}{4}]^{th}~item=Size~of~50^{th}~item

Q1 lies in the group 10-15.

l1 = 10, c.f. = 24, f = 30, i = 5

Q_1=l_1+\frac{\frac{N}{4}-c.f.}{f}\times i

Q_1=10+\frac{\frac{200}{4}-24}{30}\times 5

Q1 = 14.33

Q_3=Size~of~[\frac{3N}{4}]^{th}~item=Size~of~(\frac{3\times200}{4})^{th}~item=Size~of~150^{th}~item

Q3 lies in the group 20-25.

l1 = 20, c.f. = 100, f = 60, i = 5

Q_3=l_1+\frac{\frac{3N}{4}-c.f.}{f}\times i

Q_3=20+\frac{\frac{3\times200}{4}-100}{60}\times 5

Q3 = 24.16

Interquartile Range = Q3 – Q1 = 24.16 – 14.33 = 9.83

Quartile Deviation\frac{Q_3-Q_1}{2}=\frac{24.16-14.33}{2}=4.915

Coefficient of Quartile Deviation = \frac{Q_3-Q_1}{Q_3+Q_1}=\frac{24.16-14.33}{24.16+14.33}=0.25

Example 2:

Calculate the interquartile range, quartile deviation, and coefficient of quartile deviation from the following figures:

Quartile Deviation in Continuous Series

 

Solution:

Quartile Deviation in Continuous Series

 

Q_1=Size~of~[\frac{N}{4}]^{th}~item=Size~of~[\frac{80}{4}]^{th}~item=Size~of~20^{th}~item

Q1 lies in the group 20-30.

l1 = 20, c.f. = 14, f = 20, i = 10

Q_1=l_1+\frac{\frac{N}{4}-c.f.}{f}\times i

Q_1=20+\frac{\frac{80}{4}-14}{20}\times 10

Q1 = 23

Q_3=Size~of~[\frac{3N}{4}]^{th}~item=Size~of~(\frac{3\times80}{4})^{th}~item=Size~of~60^{th}~item

Q3 lies in the group 30-40.

l1 = 30, c.f. = 34, f = 28, i = 10

Q_3=l_1+\frac{\frac{3N}{4}-c.f.}{f}\times i

Q_3=30+\frac{\frac{3\times80}{4}-34}{28}\times 10

Q3 = 39.28

Interquartile Range = Q3 – Q1 = 39.28 – 23 = 16.28

Quartile Deviation\frac{Q_3-Q_1}{2}=\frac{39.28-23}{2}=8.14

Coefficient of Quartile Deviation\frac{Q_3-Q_1}{Q_3+Q_1}=\frac{39.28-23}{39.28+23}=0.26

Example 3:

Calculate the interquartile range, quartile deviation, and coefficient of quartile deviation from the following figures:

Quartile Deviation in Continuous Series

 

Solution:

Quartile Deviation in Continuous Series

 

Q_1=Size~of~[\frac{N}{4}]^{th}~item=Size~of~[\frac{23}{4}]^{th}~item=Size~of~5.75^{th}~item

Q1 lies in the group 10-20.

l1 = 10, c.f. = 4, f = 3, i = 10

Q_1=l_1+\frac{\frac{N}{4}-c.f.}{f}\times i

Q_1=10+\frac{\frac{23}{4}-4}{3}\times 10

Q1 = 15.83

Q_3=Size~of~[\frac{3N}{4}]^{th}~item=Size~of~(\frac{3\times23}{4})^{th}~item=Size~of~17.25^{th}~item

Q3 lies in the group 40-50.

l1 = 40, c.f. = 13, f = 8, i = 10

Q_3=l_1+\frac{\frac{3N}{4}-c.f.}{f}\times i

Q_3=40+\frac{\frac{3\times23}{4}-13}{8}\times 10

Q3 = 45.31

Interquartile Range = Q3 – Q1 = 45.31 – 15.83 = 29.48

Quartile Deviation\frac{Q_3-Q_1}{2}=\frac{45.31-15.83}{2}=14.74

Coefficient of Quartile Deviation\frac{Q_3-Q_1}{Q_3+Q_1}=\frac{45.31-15.83}{45.31+15.83}=\frac{29.48}{61.14}=0.48

Example 4:

Calculate the interquartile range, quartile deviation, and coefficient of quartile deviation from the following figures:

Quartile Deviation in Continuous Series

 

Solution:

Quartile Deviation in Continuous Series

 

Q_1=Size~of~[\frac{N}{4}]^{th}~item=Size~of~[\frac{16}{4}]^{th}~item=Size~of~4^{th}~item

Q1 lies in the group 8.5-13.5.

l1 = 8.5, c.f. = 3, f = 4, i = 5

Q_1=l_1+\frac{\frac{N}{4}-c.f.}{f}\times i

Q_1=8.5+\frac{\frac{16}{4}-3}{4}\times 5

Q1 = 9.75

Q_3=Size~of~[\frac{3N}{4}]^{th}~item=Size~of~(\frac{3\times16}{4})^{th}~item=Size~of~12^{th}~item

Q3 lies in the group 18.5-23.5.

l1 = 18.5, c.f. = 10, f = 2, i = 5

Q_3=l_1+\frac{\frac{3N}{4}-c.f.}{f}\times i

Q_3=18.5+\frac{\frac{3\times16}{4}-10}{2}\times 5

Q3 = 23.5

Interquartile Range = Q3 – Q1 = 23.5 – 9.75 = 13.75

Quartile Deviation\frac{Q_3-Q_1}{2}=\frac{23.5-9.75}{2}=6.875

Coefficient of Quartile Deviation \frac{Q_3-Q_1}{Q_3+Q_1}=\frac{23.5-9.75}{23.5+9.75}=0.41



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