Question

The mean of the following frequency data is 42, Find the missing frequencies x and y if the sum of frequencies is 100

Class  interval	0-10	10-20	20-30	30-40	40-50	50-60	60-70	70-80
Frequency	7	10	x	13	y	10	14	9

Find x and y.

Answer 1

The given data is shown as follows:

Class interval	Frequency `(f_i)`	Class mark `(x_i)`	`f_ix_i`
0-10	7	5	35
10-20	10	15	150
20-30	x	25	25x
30-40	13	35	455
40-50	y	45	45y
50-60	10	55	550
60-70	14	65	910
70-80	9	75	675
Total	`sum f_i = 63 + x+y`		`sum f_i x_i = 2775 + 25x+45y`

Sum of the frequencies = 100

⇒ Σ𝑖 𝑓𝑖 = 100
⇒ 63 + x + y = 100
⇒ x + y = 100 – 63
⇒ x + y = 37
⇒ y = 37 – x           ……..(1)
Now, the mean of the given data is given by

x = ` (sum _(i) f_i x_i )/(sum_(i) f_i )`

⇒ 42 =`(2775+25x+45y)/100`
⇒ 4200 = 2775 + 25x + 45y
⇒ 4200 – 2775 = 25x + 45y
⇒ 1425 = 25x + 45(37 – x)          [from (1)]
⇒ 1425 = 25x + 1665 – 45x
⇒ 20x = 1665 – 1425
⇒ 20x = 240
⇒ x = 12
If x = 12, then y = 37 – 12 = 25
Thus, the value of x is 12 and y is 25.