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Statistics - Quartile Deviation
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 |
|---|---|
| 9.3-9.7 | 22 |
| 9.8-10.2 | 55 |
| 10.3-10.7 | 12 |
| 10.8-11.2 | 17 |
| 11.3-11.7 | 14 |
| 11.8-12.2 | 66 |
| 12.3-12.7 | 33 |
| 12.8-13.2 | 11 |
Solution:
| Maximum Load (short-tons) | Number of Cables (f) | Class Bounderies | Cumulative Frequencies |
|---|---|---|---|
| 9.3-9.7 | 2 | 9.25-9.75 | 2 |
| 9.8-10.2 | 5 | 9.75-10.25 | 2 + 5 = 7 |
| 10.3-10.7 | 12 | 10.25-10.75 | 7 + 12 = 19 |
| 10.8-11.2 | 17 | 10.75-11.25 | 19 + 17 = 36 |
| 11.3-11.7 | 14 | 11.25-11.75 | 36 + 14 = 50 |
| 11.8-12.2 | 6 | 11.75-12.25 | 50 + 6 = 56 |
| 12.3-12.7 | 3 | 12.25-12.75 | 56 + 3 = 59 |
| 12.8-13.2 | 1 | 12.75-13.25 | 59 + 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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