Question

The following table gives the daily income of 50 workers of a factory:

Daily income (in Rs)	100 – 120	120 – 140	140 – 160	160 – 180	180 – 200
No. of workers	12	14	8	6	10

Find the mean, median and mode of the above data.

Answer 1

We have the following:

Daily income	Mid value `(x_i)`	Frequency `(f_i)`	Cumulative frequency	`(f_i × x_i)`
100 – 120	110	12	12	1320
120 – 140	130	14	26	1820
140 – 160	150	8	34	1200
160 – 180	170	6	40	1020
180 – 200	190	10	50	1900
		`Ʃ f_i` = 50		`Ʃ f_i × x_i` = 7260

Mean, x = `(Ʃ_ i  f_i × x_i)/(Ʃ _i f_i)`
              =`7260/50`

              = 145.2
 Now, N = 50

⇒ `N/2 =25`

The cumulative frequency just greater than 25 is 26 and the corresponding class is 120 – 140.
Thus, the median class is 120 – 140.
∴ l = 120, h = 20, f = 14, c = cf of preceding class = 12 and  `N/2 =25`

Now,
Median,` M_e = l + {h× ((N/2−c)/
f)}`

                    ` = 120 + {20 × ((25−12)/14)}`
                      `= (120 + 20 × 13/14)`
                       = 138.57
Mode = 3(median) – 2(mean)
                      = 3 × 138.57 – 2 × 145.2
                      = 125.31