1. Got it! Here's the cleaned-up version without the bold formatting: Determine the mode from the following frequency distribution: | Item Size Number | 4–8 | 8–12 | 12–16 | 16–20 | 20–24 | 24–28 | 28–32 | |------------------|-----|------|-------|-------|-------|-------|-------| | Frequency | 9 | 10 | 18 | 14 | 10 | 6 | 3 |

2. From the following cumulative frequency distribution table, construct a frequency distribution table and determine the **mode** of the data: | Class Boundary | Less than 10 | Less than 20 | Less than 30 | Less than 40 | Less than 50 | Less than 60 | Less than 70 | Less than 80 | |--------------------|--------------|--------------|--------------|--------------|--------------|--------------|--------------|--------------| | Cumulative Frequency | 4 | 16 | 40 | 76 | 96 | 112 | 120 | 125 |

3. Let's find the mode from the following frequency distribution: | Class Interval | 0–5 | 5–10 | 10–15 | 15–20 | 20–25 | 25–30 | 30–35 | |----------------|-----|------|-------|-------|-------|-------|-------| | Frequency | 5 | 12 | 18 | 28 | 17 | 12 | 8 |

4. Draw a greater cumulative frequency ogive using the following statistical distribution and determine the median from it: | Class Interval | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 | 50–60 | |---------------------|------|--------|--------|--------|--------|--------| | Frequency | 8 | 12 | 21 | 30 | 22 | 7 |

5. Calculate the average (mean) from the following frequency distribution table: | Class Interval | 45–54 | 55–64 | 65–74 | 75–84 | 85–94 | 95–104 | |----------------|-------|-------|-------|-------|-------|--------| | Frequency | 8 | 13 | 19 | 32 | 12 | 6 |

6. Calculate the mean using the assumed mean method (step-deviation method) from the following data: | Class Interval | 0–30 | 30–60 | 60–90 | 90–120 | 120–150 | |----------------|------|--------|--------|---------|----------| | Frequency | 12 | 15 | 20 | 25 | 8 |

7. Find the mode of the following frequency distribution: | Class Interval | 0–5 | 5–10 | 10–15 | |----------------|-----|------|--------| | Frequency | 5 | 12 | 18 | | Class Interval | 15–20 | 20–25 | 25–30 | 30–35 | |----------------|--------|--------|--------|--------| | Frequency | 28 | 17 | 12 | 8 |

8. From the following frequency distribution table of candidates' ages in an entrance examination, find the **mode**: | Age (in years) | 16–18 | 18–20 | 20–22 | 22–24 | 24–26 | |----------------|--------|--------|--------|--------|--------| | Number of Candidates | 45 | 75 | 38 | 22 | 20 |

9. Find the median from the following grouped frequency distribution: | Class Interval | Frequency | |----------------|-----------| | 0–10 | 4 | | 10–20 | 7 | | 20–30 | 10 | | 30–40 | 15 | | 40–50 | 10 | | 50–60 | 8 | | 60–70 | 5 |

10. Determine the mode of the following frequency distribution: | Class Interval | 0-5 | 5-10 | 10-15 | 15-20 | 20-25 | 25-30 | 30-35 | |---------------|------|------|------|------|------|------|------| | Frequency | 5 | 12 | 18 | 28 | 17 | 12 | 8 |

11. Find the median from the following frequency distribution: | Number Range (\(x_i\)) | 0–4 | 5–9 | 10–14 | 15–19 | 20–24 | 25–29 | 30–34 | 35–39 | |------------------------|-----|-----|-------|-------|-------|-------|-------|-------| | Frequency (\(f_i\)) | 4 | 5 | 7 | 8 | 7 | 5 | 6 | 3 |

12. Find the median of the following frequency distribution: | Class Interval | 1–5 | 6–10 | 11–15 | 16–20 | 21–25 | 26–30 | 31–35 | |----------------|-----|------|--------|--------|--------|--------|--------| | Frequency | 2 | 3 | 6 | 7 | 5 | 4 | 3 |

13. Given that the combined (weighted) mean of the following frequency distribution is 50 and the total frequency is 120, find the values of \( f_1 \) and \( f_2 \). | Class Interval | 0–20 | 20–40 | 40–60 | 60–80 | 80–100 | |----------------|------|--------|--------|--------|---------| | Frequency | 17 | \( f_1 \) | 32 | \( f_2 \) | 19 |

14. From the following frequency distribution of candidates' ages in an entrance examination, find the median: | Class Interval | 1–5 | 6–10 | 11–15 | 16–20 | 21–25 | 26–30 | 31–35 | |----------------|-----|------|-------|-------|-------|-------|-------| | Frequency | 2 | 3 | 6 | 7 | 5 | 4 | 3 |

15. **English Translation:** Draw the **less than ogive** (cumulative frequency curve) for the following frequency distribution: | Marks Obtained | 50–60 | 60–70 | 70–80 | 80–90 | 90–100 | |----------------|-------|-------|-------|-------|--------| | Frequency | 4 | 8 | 12 | 6 | 10 |

16. Let's calculate the average from the following data: | Class Interval | 5–14 | 15–24 | 25–34 | 35–44 | 45–54 | 55–64 | |----------------|------|-------|-------|-------|-------|-------| | Frequency | 3 | 6 | 18 | 20 | 10 | 3 |

17. Given that the combined mean of the following distribution is 50 and the total frequency is 120, find the values of \(f_1\) and \(f_2\): | Class Interval | 0–20 | 20–40 | 40–60 | 60–80 | 80–100 | |----------------|------|--------|--------|--------|---------| | Frequency | 17 | \(f_1\) | 32 | \(f_2\) | 19 |

18. Find the mode of the following frequency distribution: | Class Interval | 50–59 | 60–69 | 70–79 | 80–89 | 90–99 | 100–109 | |----------------|--------|--------|--------|--------|--------|----------| | Frequency | 5 | 20 | 40 | 50 | 30 | 6 |

19. Let's determine the mode from the frequency distribution below: | Class Interval | 45–54 | 55–64 | 65–74 | 75–84 | 85–94 | 95–104 | |--------------------|--------|--------|--------|--------|--------|---------| | Frequency | 4 | 6 | 40 | 76 | 96 | 112 |

20. Determine the mode from the following frequency distribution table: | Class Interval | 0–50 | 50–100 | 100–150 | 150–200 | 200–250 | 250–300 | |----------------|------|--------|---------|---------|---------|---------| | Frequency | 10 | 16 | 28 | 22 | 18 | 6 |

21. If the median of the following frequency distribution table is 27, then find the value of \(a\): | Class Interval | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 | |----------------|------|--------|--------|--------|--------| | Frequency | 3 | a | 20 | 12 | 7 |

22. Find the median from the following data: | Class Interval | 5–10 | 10–15 | 15–20 | 20–25 | 25–30 | 30–35 | 35–40 | 40–45 | |----------------|------|-------|-------|-------|-------|-------|-------|-------| | Frequency | 5 | 6 | 15 | 10 | 5 | 4 | 3 | 2 |

23. Calculate the average marks obtained by the students, given the following cumulative frequency distribution: | Marks | Number of Students | |--------------|--------------------| | Less than 10 | 6 | | Less than 20 | 10 | | Less than 30 | 18 | | Less than 40 | 30 | | Less than 50 | 46 |

24. **Find the average (mean) of the following data:** | Class Interval | 5–14 | 15–24 | 25–34 | 35–44 | 45–54 | 55–64 | |----------------|------|-------|-------|-------|-------|-------| | Frequency | 3 | 6 | 18 | 20 | 10 | 3 |

25. Prepare a greater-than cumulative frequency table from the following frequency distribution and draw an ogive on graph paper. | Class Interval | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 | 50–60 | |----------------|------|--------|--------|--------|--------|--------| | Frequency | 7 | 10 | 23 | 50 | 6 | 4 |

26. Find the median of the following frequency distribution: | Class Interval | 0–10 | 10–30 | 30–60 | 60–70 | 70–90 | |----------------|------|-------|-------|-------|-------| | Frequency | 15 | 25 | 30 | 4 | 10 |

27. If the mean of the frequency distribution is 62.8 and the total frequency is 50, then what are the values of \(x\) and \(y\) given the following table: | Class | 0–20 | 20–40 | 40–60 | 60–80 | 80–100 | 100–120 | |-------------|------|--------|--------|--------|---------|----------| | Frequency | 5 | \(x+y\) | 10 | \(y−x\) | 7 | 8 |

28. If the median of the following data is 32 and the total frequency is 100, find the values of \(x\) and \(y\): | Class Interval | 0–10 | 10–20 | 20–30 | |----------------|------|--------|--------| | Frequency | 10 | \(x\) | 25 | | Class Interval | 30–40 | 40–50 | 50–60 | |----------------|--------|--------|--------| | Frequency | 30 | \(y\) | 10 |

29. If the combined (weighted) mean of the following frequency distribution table is 54, then find the value of K: | Class Interval | 0–20 | 20–40 | 40–60 | 60–80 | 80–100 | |----------------|------|-------|-------|-------|--------| | Frequency | 7 | 11 | K | 9 | 13 |

30. Using the direct method, calculate the mean of the following data: | Class Interval | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 | |----------------|------|-------|-------|-------|-------| | Frequency | 4 | 6 | 10 | 6 | 4 |