Question

Find the quartile deviation and coefficient of quartile from the given data

$\begin{array}{cccccccc}\text{Class Interval}& 0-10& 10-20& 20-30& 30-40& 40-50& 50-60& 60-70\\ \text{Frequency}& 28& 31& 18& 21& 36& 49& 20\end{array}$

Answer 1

Calculation of Quartile Deviation and Coefficient of Quartile Deviation

$\begin{array}{ccc}\text{Class Interval}& \text{Frequency (f)}& \text{Cumulative Frequency (cf)}\\ 0-10& 28& 28\\ 10-20& 31& 59\\ 20-30& 18& 77\\ 30-40& 21& 98\\ 40-50& 36& 134\\ 50-60& 49& 183\\ 60-70& 20& 203\\ & N=203& \end{array}$

$Q_1$ = Size of $\left(\frac{N}{4}\right)$th item

= Size of $\left(\frac{203}{4}\right)$th item = 50.75th item

50.75th item lies in group 10 - 20 and falls within 59th cumulative frequency.

$Q_1=l_1+\frac{\frac{N}{4}-cf}{f}\times i$

$=10+\frac{50.75-28}{31}\times 10=10+\frac{22.75}{31}\times 10$

$=10+\frac{227.75}{31}=10+7.3=17.3$

$Q_3$ = size of $3\left(\frac{N}{4}\right)$th item

= Size of $\left(\frac{3\times 203}{4}\right)$th item

= Size of $\left(\frac{609}{4}\right)$th item

= 152.25 th item

lies in group 50 - 60 and falls within 183rd cummulative frequency.

$Q_3=l_1+\frac{3\frac{N}{4}-cf}{f}\times i$

$=50+\frac{152.25-134}{49}\times 10$

$=50+\frac{18.25}{49}\times 10=50+3.7=53.7$

Now, $QD=\frac{Q_3-Q_1}{2}$

$=\frac{53.7-17.3}{2}=\frac{36.4}{2}=18.2$

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

$=\frac{53.7-17.3}{53.7+17.3}=\frac{36.4}{71}=0.51$