Question

The following table gives the distribution of the life time of 400 neon lamps:

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

Find the median life time of a lamp.

Answer 1

Based on the given information, we can prepare the table shown above.

Here, we have, $n=400$

$\Rightarrow \frac{n}{2}=200$

The cumulative frequency just greater than $\frac{n}{2}$ is $216$ and the corresponding class is $3000–3500$.

Thus, $3000–3500$ is the median class such that

$\frac{n}{2}=200,l=3000,cf=130,f=86$, and $h=500$

We know that,

Median, $M=l+\left(\frac{\frac{n}{2}-cf}{f}\right)\times h$

Where, $l\to$ lower limit of the median class

$n\to$ total number of observations $(\sum f)$

$cf\to$ cumulative frequency of the class preceding the median class

$f\to$ frequency of the median class

$h\to$ class width

Substituting the corresponding values in the formula, we get:

$M=3000+\left(\frac{200-130}{86}\right)\times 500$

$\Rightarrow M=3000+\frac{70}{86}\times 500=3000+406.98=3406.98$

Hence, median life time of a lamp is $3406.98$ hours.