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Interquartile Range and Quartile Deviation

 1 year ago
source link: https://www.geeksforgeeks.org/interquartile-range-and-quartile-deviation/
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The extent to which the values of a distribution differ from the average of that distribution is known as Dispersion. The measures of dispersion can be either absolute or relative. The Measures of Absolute Dispersion consist of Range, Quartile Deviation, Mean Deviation, Standard Deviation, and Lorenz Curve.

What is Interquartile Range?

Range only takes the two extreme values (largest and smallest) into consideration; therefore, it is a crude measure of dispersion. This effect of extreme values on range can be avoided by using the measure of interquartile range. The difference between the values of two quartiles is known as Interquartile Range. The formula for determining Interquartile Range is as follows:

Inter Quartile Range = Q3 – Q1

Example:

Calculate Interquartile Range of the following data:

150, 110, 200, 300, 180, 320

Solution:

IQR.png

Q1 = Size~of~[\frac{N+1}{4}]^{th}~item=Size~of~[\frac{6+1}{4}]^{th}~item=Size~of~1.75^{th}~item

Size of 1.75th item = Size of 1st item + 0.75(Size of 2nd item – Size of 1st item)

Q1 = 110 + 0.75(150 – 110)

Q1 = 110 + 30

Q1 = 140

Q3 = Size~of~3[\frac{N+1}{4}]^{th}~item=Size~of~3[\frac{6+1}{4}]^{th}~item=Size~of~5.25^{th}~item

Size of 5.25th item = Size of 5th item + 0.25(Size of 6th item – Size of 5th item)

Q3 = 300 + 0.25(320 – 300)

Q3 = 300 + 5

Q3 = 305

Interquartile Range = Q3 – Q1 = 305 – 140 

Interquartile Range = 165

What is Quartile Deviation?

Quartile Deviation or Semi-Interquartile Range is the half of difference between the Upper Quartile (Q3) and the Lower Quartile (Q1). In simple terms, QD is the half of inter-quartile range. Hence, the formula for determining Quartile Deviation is as follows:

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)

Example:

Calculate Quartile Deviation of the following data:

150, 110, 200, 300, 180, 320

Solution:

IQR.png

Q1 = Size~of~[\frac{N+1}{4}]^{th}~item=Size~of~[\frac{6+1}{4}]^{th}~item=Size~of~1.75^{th}~item

Size of 1.75th item = Size of 1st item + 0.75(Size of 2nd item – Size of 1st item)

Q1 = 110 + 0.75(150 – 110)

Q1 = 110 + 30

Q1 = 140

Q3 = Size~of~3[\frac{N+1}{4}]^{th}~item=Size~of~3[\frac{6+1}{4}]^{th}~item=Size~of~5.25^{th}~item

Size of 5.25th item = Size of 5th item + 0.25(Size of 6th item – Size of 5th item)

Q3 = 300 + 0.25(320 – 300)

Q3 = 300 + 5

Q3 = 305

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

=\frac{305-140}{2}

Quartile Deviation = 82.5

What is Coefficient of Quartile Deviation?

As Quartile Deviation is an absolute measure of dispersion, one cannot use it for comparing the variability of two or more distributions when they are expressed in different units. Therefore, in order to compare the variability of two or more series with different units it is essential to determine the relative measure of Quartile Deviation, which is also known as the Coefficient of Quartile Deviation. It is studied to make the comparison between the degree of variation in different series. The formula for determining Coefficient of Quartile Deviation is as follows:

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)

Example:

Calculate Coefficient of Quartile Deviation of the following data:

150, 110, 200, 300, 180, 320

Solution:

IQR.png

Q1 = Size~of~[\frac{N+1}{4}]^{th}~item=Size~of~[\frac{6+1}{4}]^{th}~item=Size~of~1.75^{th}~item

Size of 1.75th item = Size of 1st item + 0.75(Size of 2nd item – Size of 1st item)

Q1 = 110 + 0.75(150 – 110)

Q1 = 110 + 30

Q1 = 140

Q3 = Size~of~3[\frac{N+1}{4}]^{th}~item=Size~of~3[\frac{6+1}{4}]^{th}~item=Size~of~5.25^{th}~item

Size of 5.25th item = Size of 5th item + 0.25(Size of 6th item – Size of 5th item)

Q3 = 300 + 0.25(320 – 300)

Q3 = 300 + 5

Q3 = 305

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

=\frac{305-140}{305+140}

=\frac{165}{445}

Coefficient of Quartile Deviation = 0.37

Note: The Coefficient of Quartile Deviation is calculated for making a comparison between the degree of variation in the given series. 


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