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.
The median life time of a lamp is 3406.98 hours.
Explanation
This is a standard NCERT Statistics problem requiring the median formula for grouped data. The student must first calculate cumulative frequencies to locate the median class, then apply the formula Median = l + [(N/2 - cf)/f] × h. The median class is identified where cumulative frequency first exceeds N/2 = 200.
Solution Steps
Step 1: Calculate cumulative frequencies. | Life time | f | cf | | 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 |
Step 2: Find N/2. Total lamps N = 400. N/2 = 200.
Step 3: Identify median class. The cumulative frequency just greater than 200 is 216, so median class is 3000 - 3500.
Step 4: Apply median formula. Median = l + [(N/2 - cf)/f] × h where l = 3000, N/2 = 200, cf = 130 (preceding class), f = 86, h = 500.
Step 5: Calculate. Median = 3000 + [(200 - 130)/86] × 500 = 3000 + (70/86) × 500 = 3000 + 406.98 = 3406.98 hours.