Calculate Lower Quartile and Upper Quartile from the following data :

$\begin{matrix} X & 0 - 10 & 10 - 20 & 20 - 30 & 30 - 40 & 40 - 50 & 50 - 60 & 60 - 70 & 70 - 80 f & 15 & 20 & 25 & 24 & 12 & 31 & 71 & 52 \end{matrix}$

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Solution

To calculate Lower Quartile and Upper Quartile

$\begin{matrix} X & f & c . f 0 - 10 & 15 & 15 10 - 20 & 20 & 35 20 - 30 & 25 & 60 30 - 40 & 24 & 84 40 - 50 & 12 & 96 50 - 60 & 31 & 127 60 - 70 & 71 & 198 70 - 80 & 52 & 250 \end{matrix}$

Lower Quartile $= Q_{1}$

$\frac{N}{4} = \frac{250}{4} = 62.5 t h item$

$Q_{1}$ lies in class 30 - 40

$Q_{1} = l_{1} + \frac{\frac{N}{4} - c . f .}{f} \times i$

$Q_{1} = 30 + \frac{\frac{250}{4} - 60}{24} \times 10 = 31.04$

Upper Quartile $= Q_{3}$

$\frac{3 N}{4} = \frac{3 (250)}{4} = 187.5 t h item$

$Q_{3}$ lies in class 60 - 70

$Q_{3} = l_{1} + \frac{\frac{3 N}{4} - c . f}{f} \times i$

$Q_{3} = 30 + \frac{\frac{3 \times 250}{4} - 127}{71} \times 10 = 68.52$

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