Question

Calculate the quartile devaition and it's cofficient from given data class interval 30-32, 32-34, 34-36, 36 38, 38-40, 40 42 42-44 and freuqncy 12,18,16,14,12,8,6 plz answer this question tomorrow is my exam

Answer 1

$\boxed{\begin{array}{cccc}\bf Class\: interval&\bf Frequency\: (f) &\bf C.F\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad\qquad}{}\\\sf 30 - 32&\sf 12&\sf 12\\\\\sf 32 - 34 &\sf 18&\sf 30\\\\\sf 34 - 36&\sf 16&\sf 46\\\\\sf 36 - 38&\sf 14&\sf 60\\\\\sf 38 - 40&\sf 12&\sf 72\\\\\sf 40 - 42&\sf 8&\sf 80\\\\\sf 42 - 44&\sf 6&\sf 86\end{array}}$

$\implies \sf \: Q_{1} = Size \: of \: \dfrac{N}{4}th \: item \\ \sf \: = Size \: of \: \dfrac{86}{4} th \: item \\ \sf \: = Size \: of \: 21.5th \: item$

So, 21.5th item lies in the class 32-34.

Then,

$\sf \: Q_{1} = l_{1} + \dfrac{ \dfrac{N}{4} - cf }{f} \times i \\ \\ \sf = 32 + \dfrac{21.5 - 12}{18} \times 2 \\ \\ \sf \: = 32 + \dfrac{9.5}{18} \times 2 \\ \\ \sf \: = 32 + 1.05 \\ \\ \sf = 33.05$

$\sf \implies\: Q_{3} = Size \: of \: 3( \dfrac{n}{4})th \: item \\ \sf \: = Size \: of \: 3( \dfrac{86}{4})th \: item \\ = \sf \: Size \: of \: 64.5th \: item$

So, 64.5th item lies in the 38-40 class.

Then,

$\sf \longmapsto \: Q_{ 3} = l_{1} + \dfrac{3 (\dfrac{n}{4}) - cf }{f} \times i \\ \\ \sf \: = 38 + \dfrac{3 (\dfrac{86}{4} ) - 60}{12} \times 2 \\ \\ \sf \: = 38 + \dfrac{64.5 - 60}{12} \times 2 \\ \\ = \sf \: 38 + \dfrac{4.5}{12} \times 2 \\ \\ \sf \: = 38 + 0.75 \\ \\ \sf \: = 38.75$

$\sf \longmapsto\: QD = \dfrac{ Q_{3} -Q_{1} }{2} \\ \\ = \dfrac{38.75 - 33.0}{2} \\ \\ \sf \: = 2.85$

$\sf \longmapsto \: Coefficient \: of \: QD = \dfrac{ Q_{3} -Q_{1} }{Q_{3} + Q_{1}} \\ \\ \sf \: = \dfrac{38.75 - 33.05}{38.75 + 33.05} \\ \\ \sf \: \large \boxed{\purple{= 0.079}}$