12. Fill the unknown entries \(a,b,c,d,e,f\) in the following cumulative table:
| Height (cm) | Frequency | Cumulative frequency |
|---|---|---|
| 150–155 | 12 | a |
| 155–160 | b | 25 |
| 160–165 | 10 | c |
| 165–170 | d | 43 |
| 170–175 | e | 48 |
| 175–180 | 2 | f |
| Total | 50 |
a=12, b=13, c=35, d=8, e=5, f=50
Step 1: Understand the terms
- Frequency: how many students are in that height group.
- Cumulative frequency: the total number of students from the first group up to the current group.
Step 2: First row
Frequency = 12. Since this is the first row, the cumulative frequency is the same:
\(a = 12\).
Step 3: Second row
Cumulative frequency given = 25. This means:
(First frequency) + (Second frequency) = 25.
So, \(12 + b = 25 \implies b = 13\).
Step 4: Third row
Frequency = 10. The cumulative frequency = previous cumulative (25) + 10 = 35.
So, \(c = 35\).
Step 5: Fourth row
Cumulative frequency given = 43. This means:
previous cumulative (35) + d = 43.
So, \(d = 43 - 35 = 8\).
Step 6: Fifth row
Cumulative frequency given = 48. This means:
previous cumulative (43) + e = 48.
So, \(e = 48 - 43 = 5\).
Step 7: Sixth row
Frequency = 2. The final cumulative frequency = previous cumulative (48) + 2 = 50.
So, \(f = 50\).
Final Answer: a = 12, b = 13, c = 35, d = 8, e = 5, f = 50.