Question

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

Life time (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

Find the median life time of a lamp.

Answer 1

The cumulative frequencies and their corresponding class intervals are presented below.

Life time (in hours)	Number of lamps(\(\bf{f_i}\))	Cumulative frequency
1500 - 2000	14	14
2000 - 2500	56	70
2500 - 3000	60	130
3000 - 3500	86	216
3500 - 4000	74	290
4000 - 4500	62	352
4500 - 5000	48	400
Total(n)	400

The cumulative frequency immediately greater than \(\frac{n}{2}\) (which is \(\frac{400}{2} = 200\)) is 216, corresponding to the class interval 3000 - 3500.
The median class is 3000 - 3500.
The lower limit (\(l\)) of the median class is 3000.
The frequency (\(f\)) of the median class is 86.
The cumulative frequency (\(cf\)) of the median class is 130.
The class size (\(h\)) is 500.

The median is calculated as: \(l + (\frac{\frac{n}{2} - cf}{f} \times h)\)

Median = \(3000 + (\frac{200 - 130}{86} \times 500)\)

Median = 3000 + \(\frac{70 \times 500}{86}\)
Median = 3000 + 406.967
Median = 3406.967

Therefore, the median life time of the lamps is 3406.98 hours.