Question

Solve the following :

Find out P₅₀ for the following data.

Wages (in ₹) (x)	Number of workers
0-20	4
20-40	6
40-60	10
60-80	25
80-100	15

Answer 1

Wages (in ₹) (x)	Number of workers	cf
0-20	4	4
20-40	6	10
40-60	10	20
60-80	25	45
80-100	15	60
	N = 60

P₅₀ = size of 50 ${\displaystyle {\left(\frac{\text{n}}{100}\right)}^{\text{th}}}$ observation

= size of 50${\displaystyle {\left(\frac{60}{100}\right)}^{\text{th}}}$ observation

= size of 50 (0.60)^th observation

= size of 30^th observation

c.f. which is just greater than (or equal to) 30 is 45.

l = 60, f = 25, C.F. = 20, n = 60, h = 20

${\displaystyle {\text{P}}_{\text{i}}=l+\left(\frac{\frac{\text{in}}{100}-cf}{f}\right)\times h}$

P₅₀ = l + ${\displaystyle \left(\frac{\frac{50\text{n}}{100}-\text{c.f.}}{\text{f}}\right)\times \text{h}}$

P₅₀ = ${\displaystyle 60+\left(\frac{\frac{50\times 60}{100}-20}{25}\right)\times 20}$

= ${\displaystyle 60+\left(\frac{30-20}{25}\right)\times 20}$

= ${\displaystyle 60+\left(\frac{10}{25}\right)\times 20}$

= ${\displaystyle 60+\left(\frac{200}{25}\right)}$

= 60 + 8

= 68