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 with their respective class intervals are as follows.

Life time (in hours)	Number of lamps(\(\bf{f_i}\))	Cumulative frequency
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(n)	400

Cumulative frequency just greater \(\frac{n}2 ( i.e., \frac{400}2 = 200)\) than 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 (\(cf\)) of median class = 130
Class size (\(h\)) = 5

 Median = \(l + (\frac{\frac{n}2 - cf}f \times h)\)

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

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

Therefore, median life time of lamps is 3406.98 hours.