Question

The following table gives the distribution of a lifetime of $400$ neon lamps. Find the median lifetime of a lamp.

Lifetime (in hours)	Number of lamps
$1500-2000$	$14$
$2000-2500$	$56$
$2500-3000$	$60$
$3000-3500$	$86$
$3500-4000$	$74$
$4000-4500$	$62$
$4500-5000$	$48$

Answer 1

Find the median:

Let us find the cumulative frequency

Lifetime (in hours)	Number of lamps	Cumulative
$1500-2000$	$14$	$14$
$2000-2500$	$56$	$70$
$2500-3000$	$60$	$130$
$3000-3500$	$86=\mathrm{f}$	$216$
$3500-4000$	$74$	$290$
$4000-4500$	$62$	$352$
$4500-5000$	$48$	$400$
Total	$\mathrm{n}=400$

From the table, we get to know that

$$\begin{gathered} \mathrm{n}=400 \\ \Rightarrow \frac{\mathrm{n}}{2}=\frac{400}{2}=200 \end{gathered}$$

Cumulative frequency (${\mathrm{C}}_{\mathrm{f}}$) just greater than$200$is$216$, belonging to class $3000-3500$.

$\Rightarrow$Median class$=3000–3500$

$\Rightarrow$The lower limit of the median class $\left(\mathrm{l}\right)=3000$

$\Rightarrow$Cumulative frequency of class preceding median class ${\mathrm{C}}_{\mathrm{f}}=130,$

$\Rightarrow$Frequency of median class $\mathrm{f}=86$

$\Rightarrow$Class size $\mathrm{h}=500$

$\begin{array}{rcl}\mathrm{Median}& =& \mathrm{l}+\left(\frac{{\displaystyle \frac{\mathrm{n}}{2}}-{\mathrm{C}}_{\mathrm{f}}}{\mathrm{f}}\right)\times \mathrm{h}\\ & & \\ & =& 3000+\left(\frac{200-130}{86}\right)\times 500\\ & =& 3000+\left(\frac{35000}{86}\right)\\ & =& 3000+406.97\\ & =& 3406.97\end{array}$

Therefore, the median lifetime of the lamps is $3406.97\mathrm{hours}$.