Question

Answer the following questions in detail−

Calculate P₄₀ and P₇₀ for the following data.

Expenditure	10-20	20-30	30-40	40-50	50-60
No. of students	6	4	5	6	4

Answer 1

C.I.	f	C.F.
10–20	6	6
20–30	4	10
30–40	5	15
40–50	6	21
50–60	4	25
	25

$$\begin{gathered} {\mathrm{P}}_{40}=40{\left(\frac{\mathrm{N}}{100}\right)}^{\mathrm{th}}\mathrm{item} \\ =40\times {\left(\frac{25}{100}\right)}^{\mathrm{th}}\mathrm{item} \\ ={10}^{\mathrm{th}}\mathrm{items} \end{gathered}$$

10^th item falls in 10^th C.F.

$$\begin{gathered} {\mathrm{P}}_{40}=L_1+\left(\frac{L_2-L_1}{f}\right)\left(\frac{40\mathrm{N}}{100}-c.f\right) \\ {\mathrm{P}}_{40}=L_1+\frac{L_2-L_1}{4}\left(\frac{40\mathrm{N}}{100}-c.f\right) \end{gathered}$$
$$\begin{gathered} {\mathrm{P}}_{40}=20+\frac{30-20}{4}\left(\frac{1000}{100}-6\right) \\ {\mathrm{P}}_{40}=30 \\ {\mathrm{P}}_{70}=70{\left(\frac{\mathrm{N}}{100}\right)}^{\mathrm{th}}\mathrm{item} \\ =70\times {\left(\frac{25}{100}\right)}^{\mathrm{th}}\mathrm{item} \\ =17.5^{\mathrm{th}}\mathrm{item} \end{gathered}$$

17.5^th item falls in 21^st C.F.

$$\begin{gathered} {\mathrm{P}}_{70}=40+\frac{50-40}{6}\left(\frac{1750}{100}-215\right) \\ =44.16 \end{gathered}$$