Question

Answer the following questions in detail−

Calculate Q₁ and Q₃ from the following data.

Class	10-20	20-30	30-40	40-50	50-60	60-70	70-80
Freq.	10	15	25	40	35	20	5

Answer 1

C.I.	f	C.F.
10–20	10	10
20–30	15	25
30–40	25	50
40–50	40	90
50–60	35	125
60–70	20	145
70–80	5	150
	150

$Q_1={\left(\frac{\mathrm{N}}{4}\right)}^{\mathrm{th}}={\left(\frac{150}{4}\right)}^{\mathrm{th}}=57.5^{\mathrm{th}}$

It falls in the 50^th c.f

Now,

$$\begin{gathered} Q_1=30-40 \\ {Q}_1=L_1+\left(\frac{L_2-L_1}{f}\right)\left(\frac{N}{4}-cf\right) \\ {Q}_1=30+\left(\frac{40-30}{25}\right)\left(\frac{150}{4}-25\right) \end{gathered}$$

Thus, we have:

$$\begin{gathered} Q_1=35 \\ {Q}_3=3{\left(\frac{\mathrm{N}}{4}\right)}^{\mathrm{th}} \\ =3\times {\left(\frac{150}{4}\right)}^{\mathrm{th}}=112.5 \end{gathered}$$

This falls in the 125^th c.f

$$\begin{gathered} Q_3=50-60 \\ {Q}_3=L_1+\left(\frac{L_2-L_1}{f}\right)\left(\frac{3N}{4}-cf\right) \\ {Q}_3=50+\left(\frac{60-50}{35}\right)\left(\frac{3\times 150}{4}-90\right) \end{gathered}$$

Thus, we have:
Q₃ = 56.42