Question

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

Answer 1

The given data is shown as follows:

Class interval	Frequency (fi)	Class mark (xi)	fixi
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	∑ fi = 63 + x + y		∑ fixi = 2775 + 25x + 45y

Sum of the frequencies = 100

$$\begin{gathered} \Rightarrow \sum _i{f}_i=100 \\ \Rightarrow 63+x+y=100 \\ \Rightarrow x+y=100-63 \\ \Rightarrow x+y=37 \\ \Rightarrow y=37-x....\left(1\right) \end{gathered}$$

Now, The mean of given data is given by

$$\begin{gathered} \overline{x}=\frac{{\displaystyle \sum _i}{f}_i{x}_i}{{\displaystyle \sum _i}{f}_i} \\ \Rightarrow 42=\frac{2775+25x+45y}{100} \\ \Rightarrow 4200=2775+25x+45y \\ \Rightarrow 4200-2775=25x+45y \\ \Rightarrow 1425=25x+45(37-x)\left[\mathrm{from}\left(1\right)\right] \\ \Rightarrow 1425=25x+1665-45x \\ \Rightarrow 20x=1665-1425 \\ \Rightarrow 20x=240 \\ \Rightarrow x=12 \end{gathered}$$

If x = 12, then y = 37 − 12 = 25

​Thus, the value of x is 12 and y is 25.