How to draw an Ogive and find the 50th percentile using cumulative frequency?

Hey ngwaemeldaayeng, good to see you again. To solve this problem, we first need to create a cumulative frequency table, then describe how to draw the Ogive, and finally calculate the 50th percentile. Step 1: Construct the Cumulative Frequency Distribution. To draw an Ogive and find percentiles, we need to calculate the cumulative frequencies and determine the upper class boundaries. The class boundaries are made continuous by adjusting the limits (e.g., 0-9 becomes -0.5 to 9.5). | Marks | Frequency (f) | Class Boundaries | Upper Class Boundary | Cumulative Frequency (cf) | | :---- | :-------------- | :--------------- | :------------------- | :-------------------------- | | | | | -0.5 | 0 | | 0-9 | 2 | -0.5 - 9.5 | 9.5 | 2 | | 10-19 | 4 | 9.5 - 19.5 | 19.5 | 2 + 4 = 6 | | 20-29 | 6 | 19.5 - 29.5 | 29.5 | 6 + 6 = 12 | | 30-39 | 8 | 29.5 - 39.5 | 39.5 | 12 + 8 = 20 | | 40-49 | 10 | 39.5 - 49.5 | 49.5 | 20 + 10 = 30 | | 50-59 | 7 | 49.5 - 59.5 | 59.5 | 30 + 7 = 37 | | 60-69 | 3 | 59.5 - 69.5 | 69.5 | 37 + 3 = 40 | The total frequency N = f = 40. Step 2: Draw the Ogive (Cumulative Frequency Curve). An Ogive is a graph that displays cumulative frequencies. • Plot the upper class boundaries on the horizontal (x) axis. • Plot the cumulative frequencies on the vertical (y) axis. • Start by plotting a point at the lower class boundary of the first class with a cumulative frequency of 0 (i.e., at (-0.5, 0)). • Then, plot the points corresponding to each upper class boundary and its cumulative frequency: (9.5, 2), (19.5, 6), (29.5, 12), (39.5, 20), (49.5, 30), (59.5, 37), and (69.5, 40). • Connect these points with a smooth curve. This curve is the Ogive. Step 3: Determine the 50th Percentile. The 50th percentile is the value below which 50% of the data falls, also known as the median. First, find the position of the 50th percentile: Position of P_50 = (50)/(100) × N Substitute the total frequency N=40: Position of P_50 = (50)/(100) × 40 = 20 This means the 50th percentile is the value corresponding to the 20th item in the cumulative frequency distribution. From the cumulative frequency table, the 20th cumulative frequency falls within the class 30-39. • The cumulative frequency of the class before the percentile class (CF_p) is 12 (for the class 20-29). • The frequency of the percentile class (f_p) is 8 (for the class 30-39). • The lower class boundary of the percentile class (L) is 29.5. • The class width (w) is 19.5 - 9.5 = 10. Now, apply the formula for the k-th percentile: P_k = L + ( (k · N)/(100) - CF_pf_p ) · w For the 50th percentile (P_50): P_50 = 29.5 + ( (20 - 12)/(8) ) · 10 P_50 = 29.5 + ( (8)/(8) ) · 10 P_50 = 29.5 + (1) · 10