Question

Find out median in the following series:

Size (less than)	5	10	15	20	25	30	35
Frequency	1	3	13	17	27	36	38

Answer 1

Converting the Cumulative series of 'less than' type into a simple frequency distribution.

Size	Cumulative Frequency	Frequency
0 − 5 5 − 10 10 − 15 15 − 20	1 3 13 17(c.f.)	1 3 − 1 = 2 13 − 3 = 10 17 − 13 = 4
(l₁)20 − 25	27	27 − 17 = 10 (f)
25 − 30 30 − 35	36 38	36 − 27 = 9 38 − 36 = 2
		$\mathrm{\Sigma }f=N=38$

Median = Size of ${\left(\frac{N}{2}\right)}^{\mathrm{th}}$ item
or, M = Size of ${\left(\frac{38}{2}\right)}^{\mathrm{th}}$ item
or, M = Size of 19^th item

Hence, median lies in class interval 20 − 25

$$\begin{gathered} \mathrm{Median}\mathrm{or}\mathrm{M}=l_{\mathit{1}}+\frac{{\displaystyle \frac{N}{2}}\mathit{-}c\mathit{.}f\mathit{.}}{f}\mathit{\times }i \\ \mathrm{or},\mathrm{M}=20+\frac{19-17}{10}\times 5 \\ \mathrm{or},\mathrm{M}=20+\frac{2}{10}\times 5 \\ \mathrm{or},\mathrm{M}=20+\frac{10}{10} \\ \mathrm{or},\mathrm{M}=20+1 \\ \Rightarrow \mathrm{M}=21 \end{gathered}$$

Hence, the median value of the above series is 21.