1. **Problem:** Given the marks of several students, we need to draw a frequency distribution table, calculate the Interquartile Range (IQR), and the Semi-Interquartile Range (SIQR). 2. **Step 1: Organize the data into a frequency distribution table.** - The marks are: 79, 80, 49, 68, 70, 51, 41, 10, 18, 28, 19, 29, 30, 36, 33, 50, 43, 49, 41, 45, 47, 55, 55, 60, 50, 40, 31, 32, 10, 20, 44, 11, 21, 29, 30, 25, 26, 35, 33, 48, 42, 70, 51, 20, 26, 34, 35, 34, 67, 71. - Group the marks into class intervals (e.g., 0-9, 10-19, 20-29, ..., 70-79, 80-89). | Class Interval | Frequency | |---------------|-----------| | 0 - 9 | 0 | | 10 - 19 | 4 (10,10,11,18,19) actually 5 | | 20 - 29 | 7 (20,20,21,25,26,26,28,29,29) actually 9 | | 30 - 39 | 8 (30,30,31,32,33,33,34,34,35,35,36) actually 11 | | 40 - 49 | 9 (40,41,41,42,43,44,45,47,48,49,49) actually 11 | | 50 - 59 | 6 (50,50,51,51,55,55) actually 6 | | 60 - 69 | 3 (60,67,68) actually 4 (including 70?) | | 70 - 79 | 3 (70,70,71,79) actually 4 | | 80 - 89 | 1 (80) | - Correct frequencies: | Class Interval | Frequency | |---------------|-----------| | 0 - 9 | 0 | | 10 - 19 | 5 | | 20 - 29 | 9 | | 30 - 39 | 11 | | 40 - 49 | 11 | | 50 - 59 | 6 | | 60 - 69 | 4 | | 70 - 79 | 4 | | 80 - 89 | 1 | 3. **Step 2: Calculate the Interquartile Range (IQR).** - Sort the data in ascending order. - Number of data points $n=50$. - Find $Q_1$ (25th percentile) and $Q_3$ (75th percentile). - Position of $Q_1 = \frac{n+1}{4} = \frac{51}{4} = 12.75$th value. - Position of $Q_3 = 3 \times \frac{n+1}{4} = 38.25$th value. - Sorted data (first 15 values): 10,10,11,18,19,20,20,21,25,26,26,28,29,29,30 - $Q_1$ lies between 12th and 13th values: 28 and 29 - $Q_1 = 28 + 0.75 \times (29 - 28) = 28 + 0.75 = 28.75$ - Sorted data (around 38th value): 35 (36th), 36 (37th), 40 (38th), 41 (39th) - $Q_3$ lies between 38th and 39th values: 40 and 41 - $Q_3 = 40 + 0.25 \times (41 - 40) = 40 + 0.25 = 40.25$ - Calculate IQR: $$IQR = Q_3 - Q_1 = 40.25 - 28.75 = 11.5$$ 4. **Step 3: Calculate the Semi-Interquartile Range (SIQR).** $$SIQR = \frac{IQR}{2} = \frac{11.5}{2} = 5.75$$ **Final answers:** - Frequency distribution table as above. - Interquartile Range $= 11.5$ - Semi-Interquartile Range $= 5.75$