Answer to Question #95875 in Statistics and Probability for Sana

Question

Calculate the quartile deviation and it's co efficient of quartile deviation for the following items.
X: 5 12 9 15 24 18 12

Answer 1

At first we neet to sort the initial data

$$5,9,12,12,15,18,24$$

To calculate quartile deviation

$${Q_{dev}} = \frac{{{Q_{3/4}} - {Q_{1/4}}}}{2}$$

We need to calculate the 1/4 quartile and 3/4 quartile. Because the number of elements in our list is odd, the 1/4 quartile is defined as the average of the median of the $\frac{{n - 1}}{2}$ smallest elements and the median of the $\frac{{n + 1}}{2}$ smalles elements. The median of the first 3 smalles elements $5,9,12$ is $9$ and the median of the first 4 smalles elements $5,9,12,12$ is $\frac{{9 + 12}}{2} = 10.5$ . Thus ${Q_{1/4}} = \frac{{9 + 10.5}}{2} = 9.75$ . The same procedure we shall do for the 3/4 quartile (but we shall use largest instead smallest). I.e. The median of the first 3 largest elements $15,18,24$ is $18$ and The median of the first 4 largest elements $12,15,18,24$ is $\frac{{15 + 18}}{2} = 16.5$ . Thus ${Q_{3/4}} = \frac{{16.5+ 18}}{2} = 17.25$ . Now we can calculate the quartile deviation

$${Q_{dev}} = \frac{{{Q_{3/4}} - {Q_{1/4}}}}{2} = \frac{{17.25 - 9.75}}{2} = 3.75$$

And we can calculate quartive variation coefficient

$${Q_{var }} = \frac{{{Q_{3/4}} - {Q_{1/4}}}}{{{Q_{3/4}} + {Q_{1/4}}}} \cdot 100[\% ] = \frac{5}{{18}} \cdot 100[\% ] \approx 27.778[\% ]$$