Question

Question 5
The following table gives the distribution of the life time of 400 neon lamps:

$\begin{array}{cc}Lifetime\left(inhours\right)& Numberoflamps\\ 1500-2000& 14\\ 2000-2500& 56\\ 2500-3000& 60\\ 3000-3500& 86\\ 3500-4000& 74\\ 4000-4500& 62\\ 4500-5000& 48\end{array}$

Find the median life time of a lamp.

Answer 1

We can find the cumulative frequencies with their respective class intervals as below:

$\begin{array}{ccc}Lifetime& Numberoflamps\left(f_i\right)& Cumulativefrequency\\ 1500-2000& 14& 14\\ 2000-2500& 56& 14+56=70\\ 2500-3000& 60& 70+60=130\\ 3000-3500& 86& 130+86=216\\ 3500-4000& 74& 216+74=290\\ 4000-4500& 62& 290+62=352\\ 4500-5000& 48& 352+48=400\\ Total\left(n\right)& 400& \end{array}$

Now, we may observe that cumulative frequency just greater than $\frac{n}{2}(i.e.,\frac{400}{2}=200)$ is 216 belonging to class interval 3000 - 3500.

Median class = 3000 - 3500

Lower limit $l$ of median class = 3000

Frequency $f$ of median class = 86

Cumulative frequency $c_f$ of class preceding median class = 130 Class size $h$ = 500

$Median=l+\left(\frac{\frac{n}{2}-c_f}{f}\right)\times h=3000\times \left(\frac{200-130}{86}\right)\times 500=3000+\frac{70\times 500}{86}=3000+406.98=3406.98$

So, median life time of lamps is 3406.98 hours.