Statistics - Quartile Deviation


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It depends on the lower quartile ${Q_1}$ and the upper quartile ${Q_3}$. The difference ${Q_3 - Q_1}$ is called the inter quartile range. The difference ${Q_3 - Q_1}$ divided by 2 is called semi-inter quartile range or the quartile deviation.

Formula

${Q.D. = \frac{Q_3 - Q_1}{2}}$

Coefficient of Quartile Deviation

A relative measure of dispersion based on the quartile deviation is known as the coefficient of quartile deviation. It is characterized as

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

Example

Problem Statement:

Calculate the quartile deviation and coefficient of quartile deviation from the data given below:

Maximum Load
(short-tons)
Number of Cables
9.3-9.722
9.8-10.255
10.3-10.712
10.8-11.217
11.3-11.714
11.8-12.266
12.3-12.733
12.8-13.211

Solution:

Maximum Load
(short-tons)
Number of Cables
(f)
Class
Bounderies
Cumulative
Frequencies
9.3-9.729.25-9.752
9.8-10.259.75-10.252 + 5 = 7
10.3-10.71210.25-10.757 + 12 = 19
10.8-11.21710.75-11.2519 + 17 = 36
11.3-11.71411.25-11.7536 + 14 = 50
11.8-12.2611.75-12.2550 + 6 = 56
12.3-12.7312.25-12.7556 + 3 = 59
12.8-13.2112.75-13.2559 + 1 = 60

${Q_1}$

Value of ${\frac{n}{4}^{th}}$ item =Value of ${\frac{60}{4}^{th}}$ thing = ${15^{th}}$ item. Thus ${Q_1}$ lies in class 10.25-10.75.

$ {Q_1 = 1+ \frac{h}{f}(\frac{n}{4} - c) \\[7pt] \,Where\ l=10.25,\ h=0.5,\ f=12,\ \frac{n}{4}=15\ and\ c=7 , \\[7pt] \, = 10.25+\frac{0.5}{12} (15-7) , \\[7pt] \, = 10.25+0.33 , \\[7pt] \, = 10.58 }$

${Q_3}$

Value of ${\frac{3n}{4}^{th}}$ item =Value of ${\frac{3 \times 60}{4}^{th}}$ thing = ${45^{th}}$ item. Thus ${Q_3}$ lies in class 11.25-11.75.

$ {Q_3 = 1+ \frac{h}{f}(\frac{3n}{4} - c) \\[7pt] \,Where\ l=11.25,\ h=0.5,\ f=14,\ \frac{3n}{4}=45\ and\ c=36 , \\[7pt] \, = 11.25+\frac{0.5}{14} (45-36) , \\[7pt] \, = 11.25+0.32 , \\[7pt] \, = 11.57 }$

Quartile Deviation

$ {Q.D. = \frac{Q_3 - Q_1}{2} \\[7pt] \, = \frac{11.57 - 10.58}{2} , \\[7pt] \, = \frac{0.99}{2} , \\[7pt] \, = 0.495 }$

Coefficient of Quartile Deviation

${Coefficient\ of\ Quartile\ Deviation\ = \frac{Q_3 - Q_1}{Q_3 + Q_1} \\[7pt] \, = \frac{11.57 - 10.58}{11.57 + 10.58} , \\[7pt] \, = \frac{0.99}{22.15} , \\[7pt] \, = 0.045 }$


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