Question

Calculate median, given the following data:

Mid-value	20	30	40	50	60	70
Male (c.f.)	12	25	42	46	48	50

c.f. = Cumulative Frequency

Answer 1

For the calculation of median, the given mid-values must be converted into class intervals using the following formula.
$\mathrm{Value}\mathrm{of}\mathrm{adjustment}=\frac{\mathrm{Mid}-\mathrm{point}\mathrm{of}\mathrm{one}\mathrm{class}-\mathrm{Mid}-\mathrm{point}\mathrm{of}\mathrm{preceeding}\mathrm{class}}{2}$
The value obtained is then added to the mid-point to obtain the upper limit and subtracted from the mid-point to obtain the lower limit. In this manner, we obtain the following distribution.

Mid Value	Class Interval	Cumulative Frequency (c.f.)	Frequency (f)
20 30 40 50 60 70	15 − 25 25 − 35 35 − 45 45 − 55 55 − 65 65 − 75	12 (c.f.) 25 42 46 48 50	12 25 − 12 = 13 (f) 42 − 25 = 17 46 − 42 = 4 48 − 46 = 2 50 − 48 = 2
			N = ∑f = 50

Median class is given by the size of ${\left(\frac{N}{2}\right)}^{th}$ item, i.e.${\left(\frac{50}{2}\right)}^{th}$ item, which is 25^th item.
This corresponds to the class interval of (25 − 35), so this is the median class.
$$\begin{gathered} \mathrm{Median}=l_1+\frac{{\displaystyle \frac{N}{2}}-c.f.}{f}\times i \\ \mathrm{so},\mathrm{Median}=l_1+\frac{{\displaystyle \frac{50}{2}}-c.f.}{f}\times i \\ \mathrm{or},\mathrm{Median}=25+\frac{25-12}{13}\times 10 \\ \mathrm{or},\mathrm{Median}=25+\frac{130}{13} \\ \mathrm{Thus},\mathrm{Median}=25+10=35 \end{gathered}$$