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

$\begin{matrix} L i f e t i m e (i n h o u r s) & N u m b e r o f l a m p s 1500 - 2000 & 14 2000 - 2500 & 56 2500 - 3000 & 60 3000 - 3500 & 86 3500 - 4000 & 74 4000 - 4500 & 62 4500 - 5000 & 48 \end{matrix}$

Find the median life time of a lamp.

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Solution

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

$\begin{matrix} L i f e t i m e & N u m b e r o f l a m p s (f_{i}) & C u m u l a t i v e f r e q u e n c y 1500 - 2000 & 14 & 14 2000 - 2500 & 56 & 14 + 56 = 70 2500 - 3000 & 60 & 70 + 60 = 130 3000 - 3500 & 86 & 130 + 86 = 216 3500 - 4000 & 74 & 216 + 74 = 290 4000 - 4500 & 62 & 290 + 62 = 352 4500 - 5000 & 48 & 352 + 48 = 400 T o t a l (n) & 400 \end{matrix}$

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

$M e d i a n = l + (\frac{\frac{n}{2} - c_{f}}{f}) \times h = 3000 \times (\frac{200 - 130}{86}) \times 500 = 3000 + \frac{70 \times 500}{86} = 3000 + 406.98 = 3406.98$

So, median life time of lamps is 3406.98 hours.

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The following table gives the distribution of the life time of 400 neon lamps: Life time (in hours)Number of lamps1500−2000142000−2500562500−3000603000−3500863500−4000744000−4500624500−500048 Find the median life time of a lamp.