Question

Answer the following questions in detail−

From the following frequency distribution calculate first and third quartile.

x	5	4	9	12	15	6	10
f	8	6	12	8	6	9	10

Answer 1

Xi	fi	C.F.
4	6	6
5	8	14
6	9	23
9	12	35
10	10	45
12	8	53
15	6	59
	59

The quartiles can be found using the following formulas:

$$\begin{gathered} Q_1={\left(\frac{\mathrm{N}+1}{4}\right)}^{\mathrm{th}}\mathrm{item} \\ =\left(\frac{59+1}{4}\right)={15}^{\mathrm{th}} \end{gathered}$$

It falls in the 23rd c.f which corresponds to item having value 6.

Thus, Q₁ = 6

$Q_3=3{\left(\frac{\mathrm{N}+1}{4}\right)}^{\mathrm{th}}\mathrm{item}=3\times \left(\frac{59+1}{4}\right)={45}^{\mathrm{th}}\mathrm{item}.$

It falls in the 45th c.f which corresponds to item having value 10.

Thus, Q₃ = 10