1. **Problem Statement:** Represent the given data using (a) Histogram, (b) Frequency polygon, (c) Frequency curve, and (d) Cumulative frequency curve (ogive). Given data: | Marks | 0-20 | 21-30 | 31-40 | 41-50 | 51-60 | 61-70 | 71-80 | |-----------|-------|-------|-------|-------|-------|-------|-------| | No. of students | 42 | 38 | 120 | 84 | 48 | 36 | 31 | 2. **Histogram:** - A histogram is a bar graph representing frequency distribution. - The x-axis represents the class intervals (marks). - The y-axis represents the frequency (number of students). - Each bar's height corresponds to the frequency of the class interval. 3. **Frequency Polygon:** - Plot the midpoints of each class interval on the x-axis. - Plot the corresponding frequencies on the y-axis. - Connect the points with straight lines. - Midpoints are calculated as $\frac{\text{lower limit} + \text{upper limit}}{2}$. Calculate midpoints: - 0-20: $\frac{0+20}{2} = 10$ - 21-30: $\frac{21+30}{2} = 25.5$ - 31-40: $\frac{31+40}{2} = 35.5$ - 41-50: $\frac{41+50}{2} = 45.5$ - 51-60: $\frac{51+60}{2} = 55.5$ - 61-70: $\frac{61+70}{2} = 65.5$ - 71-80: $\frac{71+80}{2} = 75.5$ 4. **Frequency Curve:** - Smooth curve drawn by joining the points of the frequency polygon. - It represents the continuous distribution of frequencies. 5. **Cumulative Frequency Curve (Ogive):** - Calculate cumulative frequencies by adding frequencies successively: - 42, 42+38=80, 80+120=200, 200+84=284, 284+48=332, 332+36=368, 368+31=399 - Plot cumulative frequency against the upper class boundary: - Upper boundaries: 20, 30, 40, 50, 60, 70, 80 - Join points with a smooth curve. **Summary:** - Histogram: Bars with heights 42, 38, 120, 84, 48, 36, 31 at intervals 0-20, 21-30, ..., 71-80. - Frequency polygon: Points at midpoints (10, 25.5, 35.5, 45.5, 55.5, 65.5, 75.5) with frequencies connected by lines. - Frequency curve: Smooth curve through frequency polygon points. - Ogive: Cumulative frequencies plotted against upper class boundaries connected smoothly. Final answer: The data can be represented graphically as described above.