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 |

Q.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 |

Let's first convert the cumulative frequency table into a regular frequency distribution. From the given data: - 4 students scored less than 10 → So, 4 students fall in the 0–10 range - 16 students scored less than 20 → So, (16 − 4) = 12 students fall in the 10–20 range - Similarly, we subtract successive cumulative frequencies to get the actual frequencies The frequency distribution table becomes: | Marks Range | Number of Students | |-------------|--------------------| | 0–10 | 4 | | 10–20 | 12 | | 20–30 | 24 | | 30–40 | 36 | | 40–50 | 20 | | 50–60 | 16 | | 60–70 | 8 | | 70–80 | 5 | ∴ The modal class is **30–40** (highest frequency = 36) To find the mode, we use the formula: \[ \text{Mode} = l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h \] Where: - \(l = 30\) (lower boundary of modal class) - \(f_1 = 36\) (frequency of modal class) - \(f_0 = 24\) (frequency of class before modal class) - \(f_2 = 20\) (frequency of class after modal class) - \(h = 10\) (class width) Substituting values: \[ \text{Mode} = 30 + \frac{36 - 24}{2 \times 36 - 24 - 20} \times 10 = 30 + \frac{12}{28} \times 10 = 30 + \frac{120}{28} = 30 + 4.29 = 34.29 \ (\text{approx}) \] Therefore, the mode of the data is approximately 34.29.