Question

Compute median from the following data:

Mid-values	37.5	42.5	47.5	52.5	57.5
No. of Students	30	20	15	13	22

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 Values	Class Interval	Frequency (f)	Cumulative Frequency (c.f.)
37.5 42.5	35 − 40 40 − 45	30 20(f)	30(c.f.) 50
47.5	45 − 50	15	65
52.5 57.5	50 − 55 55 − 60	13 22	78 100
		N = ∑f =100

Median class is given by the size of ${\left(\frac{N}{2}\right)}^{\mathrm{th}}$ item, i.e.${\left(\frac{100}{2}\right)}^{\mathrm{th}}$ item, which is 50^th item.
This corresponds to the class interval of (40 − 45), so this is the median class.
$$\begin{gathered} \mathrm{Median}=l_1+\frac{{\displaystyle \frac{N}{2}}-c.f.}{f}\times i \\ \mathrm{so},\mathrm{Median}=l_1+\frac{{\displaystyle \frac{100}{2}}-c.f.}{f}\times i \\ \mathrm{or},\mathrm{Median}=40+\frac{50-30}{20}\times 5 \\ \mathrm{or},\mathrm{Median}=40+5=45 \\ \mathrm{Thus},\mathrm{Median}=45 \end{gathered}$$