Question

Life of electric lamps is given in the following table. Calculate the median and the mode.

Life in hours	Number of lamps
Below 400	4
400-800	12
800-1200	40
1200-1600	41
1600-2000	27
2000-2400	13
2400-2800	9
Above 2800	4

Answer 1

Class Interval	f	c.f.
Below 400	4	4
400 – 800	12	16
800 – 1200	40	56
1200 – 1600	41	97
1600 – 2000	27	124
2000 – 2400	13	137
2400 – 2800	9	146
above 2800	4	150

Median class is given by the size of the ${\left(\frac{N}{2}\right)}^{\mathrm{th}}$ item, i.e. the ${\left(\frac{150}{2}\right)}^{\mathrm{th}}$ item, which is the 75^th item. This corresponds to the class interval 1200−1600, so this is the median class.
$$\begin{gathered} \mathrm{Median}=l_1+\frac{\left({\displaystyle \frac{N}{2}}\right)-c.f.}{f}\times i \\ =1200+\frac{75-56}{41}\times 400 \\ =1200+\frac{7600}{41} \\ =1200+185.36 \\ =1385.36 \\ \therefore \boxed{\mathrm{Median}=1385.36} \end{gathered}$$

By inspection, we can spot that (1200–1600) has the highest frequency, i.e. 41. So, this is the modal class.
$$\begin{gathered} \mathrm{Mode}=1200+\frac{41-40}{2\times 41-40-27}\times 400 \\ =1200+\frac{400}{15} \\ =1200+26.66 \\ =1226.66 \\ \therefore \boxed{\mathrm{Mode}=1226.66} \end{gathered}$$