Question

The mean of the following data is 42. Find the missing frequencies x and y if the sum of frequencies is 100.
$\begin{array}{ccccccccc}\text{Class interval}& 0-10& 10-20& 20-30& 30-40& 40-50& 50-60& 60-70& 70-80\\ \text{Frequency}& 7& 10& x& 13& y& 10& 14& 9\end{array}$

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		∑(fi×xi)=2775+25x+45y

Sum of the frequencies = 100

⇒ ∑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

$\overline{x}$ = ∑(fi×xi)/∑fi

⇒42= $\frac{2775+25x+45y}{100}$

⇒ 4200 = 2775 + 25 x + 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.