Question 1: A cumulative frequency graph for a dataset shows an S-curve. Explain what the initial, steep, and plateau sections represent in terms of data distribution.

Answer: Initial section indicates slow data accumulation, steep section shows rapid increase in data points, and plateau indicates saturation or maximum frequency reached.

Question 2: Construct a simple S-curve by plotting the following cumulative frequencies at x-values 1 to 5: [2, 5, 9, 12, 14]. Use the grid provided.

Answer: Students should plot points (1,2), (2,5), (3,9), (4,12), (5,14) and draw a smooth S-shaped curve.

Question 3: Calculate the approximate median value from a cumulative frequency table where the total frequency is 100. The median corresponds to the cumulative frequency of 50. If the cumulative frequency at x=4 is 45 and at x=5 is 60, estimate the median x-value.

Answer: Approximately 4.17 (using linear interpolation)

Question 4: Describe the key features of an S-curve in the context of student age distribution in a school.

Answer: Slow initial increase (few students of certain ages), rapid rise (most students in middle ages), then plateau as all age groups are represented.

Question 5: A data set's cumulative frequency graph is roughly symmetric with a gentle S-shape. What does this suggest about the underlying data distribution?

Answer: It suggests the data may be approximately normally distributed.

Question 6: Plot the S-curve for the following data points: (x=1, 3), (2, 7), (3, 15), (4, 20), (5, 23). Use the grid provided.

Answer: Plot points and draw a smooth S-shaped curve passing through these points.

Question 7: A cumulative frequency graph shows a very steep middle section. What might this indicate about the data set?

Answer: Most data values are concentrated in the middle range, indicating a cluster around the median.

Question 8: In a real-world scenario, how can an S-curve help in understanding population growth over time?

Answer: It can show slow initial growth, rapid expansion, and then stabilization, indicating stages of population change.

Question 9: Identify and correct the mistake: A student claims that the steepest part of an S-curve always occurs at the median of the data.

Answer: Incorrect; the steepest part occurs where the rate of increase is highest, which may not be exactly at the median.

Question 10: Challenge: Given a cumulative frequency distribution with total frequency 200, the cumulative frequency reaches 150 at x=10, and 180 at x=11. Estimate the x-value where the cumulative frequency is 170.

Answer: Approximately 10.5 (interpolated between 10 and 11)

Question 11: Mixed Review: List three key characteristics of an S-curve and explain their significance in data analysis.

Answer: Key characteristics include the initial slow rise (indicates beginning phase), rapid middle growth (indicates exponential or rapid change), and plateau (shows saturation or limit). These help identify stages of growth and distribution shape.