Question

Find the missing frequencies in the following frequency distribution whose mean is 34.

x	10	20	30	40	50	60	Total
f	4	f1	8	f2	3	4	35

Answer 1

We know that,
Mean = $\frac{{\displaystyle \sum _{}{x}_i{f}_i}}{\sum _{}{f}_i}$

For the following data:

x	10	20	30	40	50	60	Total
f	4	f1	8	f2	3	4	35

Mean = $\frac{\left(10\times 4\right)+\left(20\times f_1\right)+\left(30\times 8\right)+\left(40\times f_2\right)+\left(50\times 3\right)+\left(60\times 4\right)}{35}$
$$\begin{gathered} \Rightarrow 34=\frac{40+20f_1+240+40f_2+150+240}{35} \\ \Rightarrow 34\left(35\right)=670+20f_1+40f_2 \\ \Rightarrow 1190-670=20f_1+40f_2 \\ \Rightarrow 20f_1+40f_2=520 \\ \Rightarrow 20\left(f_1+2f_2\right)=520 \\ \Rightarrow f_1+2f_2=\frac{520}{20} \\ \Rightarrow f_1+2f_2=26 \\ \Rightarrow f_1=26-2f_2...\left(1\right) \end{gathered}$$

Also, 4 + f₁ + 8 + f₂ + 3 + 4 = 35
⇒ 19 + f₁ + f₂ = 35
⇒ f₁ + f₂ = 35 − 19
⇒ f₁ + f₂ = 16
⇒ 26 − 2f₂ + f ₂ = 16 (from (1))
⇒ 26 − f₂ = 16
⇒ 26 − 16 = f₂
⇒ f₂ = 10

Putting the value of f₂ in (1), we get
f₁ = 26 − 2(10) = 6

Hence, the value of f₁ and f₂ is 6 and 10, respectively.