Question

An Incomplete distribution is given below:

Variables	10−20	20−30	30−40	40−50	50−60	60−70	70−80	Total
Frequency	12	30	?	65	?	25	18	229

You are given that the median value is 46. Using the median formula, fill up the missing Frequency.

Answer 1

Given, Median = 46
Let the missing frequencies be f₁ & f_2.

Class Interval	Frequency (f)	Cumulative Frequency (c.f.)
10 − 20 20 − 30 30 − 40	12 30 f1	12 42 42 + f1 (c.f.)
40 − 50	65(f)	107 + f1
50 − 60 60 − 70 70 − 80	f2 25 18	107 + f1 + f2 132 + f1 + f2 229
	N = ∑f =229

Median class is given by the size of ${\left(\frac{N}{2}\right)}^{\mathrm{th}}$ item, i.e.${\left(\frac{229}{2}\right)}^{\mathrm{th}}$ item, which is 114.5^th item.
This corresponds to the class interval of (40 − 50) as median is 46.
$$\begin{gathered} \mathrm{Median}=l_1+\frac{{\displaystyle \frac{N}{2}}-c.f.}{f}\times i \\ \mathrm{so},46=40+\frac{{\displaystyle \frac{229}{2}}-\left(\mathrm{42\; +}{f}_1\right)}{65}\times 10 \\ \mathrm{or},46=40+\frac{114.5-42-f_1}{65}\times 10 \\ \mathrm{or},390=725-10f_1 \\ \mathrm{or},10f_1=335 \\ \Rightarrow f_1=33.5 \end{gathered}$$

132 + f₁ + f₂ + 18 = 229
or, f₂ = 229 − 132 − 18 − 33.5
$\Rightarrow$ f₂ = 45.5

Therefore, f₁ is 33.5 and f₂ is 45.5.