15. Weekly income of 600 families:
| Income (Rs) | 0–1000 | 1000–2000 | 2000–3000 | 3000–4000 | 4000–5000 | 5000–6000 |
|---|---|---|---|---|---|---|
| No. of families | 250 | 190 | 100 | 40 | 15 | 5 |
Compute the median income.
≈ Rs 1263.16
Step 1: First find the total number of families.
Total families = 250 + 190 + 100 + 40 + 15 + 5 = 600.
Step 2: To find the median, we need \(N/2\).
Here, \(N = 600\). So, \(N/2 = 600/2 = 300\).
Step 3: Now prepare the cumulative frequency (CF):
| Income (Rs) | No. of families (f) | Cumulative frequency (CF) |
|---|---|---|
| 0–1000 | 250 | 250 |
| 1000–2000 | 190 | 250 + 190 = 440 |
| 2000–3000 | 100 | 540 |
| 3000–4000 | 40 | 580 |
| 4000–5000 | 15 | 595 |
| 5000–6000 | 5 | 600 |
Step 4: Look for the median class.
We need the class where the 300th value lies. CF just before 300 is 250 (from 0–1000). The next CF is 440 (for 1000–2000), which covers the 300th value.
So, the median class = 1000–2000.
Step 5: Write the formula for median:
\[ \text{Median} = l + \left(\dfrac{\dfrac{N}{2} - cf}{f}\right) \times h \]
Where:
Step 6: Substitute the values:
\( \text{Median} = 1000 + \left(\dfrac{300 - 250}{190}\right) \times 1000 \)
Step 7: Simplify step by step:
\( = 1000 + \left(\dfrac{50}{190}\right) \times 1000 \)
\( = 1000 + 0.26316 \times 1000 \)
\( = 1000 + 263.16 \)
\( = 1263.16\)
Final Answer: The median income ≈ Rs 1263.16.