Question 1: Calculate the frequency density for the class interval 20-25 with a frequency of 15.

Answer: 3

Question 2: A histogram has class intervals 0-10 (width 10, frequency 20), 10-20 (width 10, frequency 30), and 20-40 (width 20, frequency 40). Calculate the height of each bar and identify if the histogram is accurately representing the data.

Answer: 0.2, 0.3, 0.2; the third bar's height is incorrect if it was plotted as 0.4 based on frequency density.

Question 3: Explain why using frequency alone can be misleading when class widths are unequal in a histogram.

Answer: Because wider classes tend to have higher frequencies, which can give a false impression of more data in those ranges. Frequency density adjusts for class width, providing a more accurate comparison.

Question 4: A histogram shows data with class intervals 30-40 (width 10, frequency 25), 40-55 (width 15, frequency 30), and 55-70 (width 15, frequency 45). Calculate the frequency density of each class.

Answer: 2.5, 2, 3

Question 5: Identify the mistake in the following statement: 'Since the class width varies, the height of the bars in the histogram should directly represent the frequency.'

Answer: The height should represent frequency density, not just frequency.

Question 6: Construct a histogram on the grid for the data: Class intervals 10-20, 20-30, 30-50 with frequencies 20, 40, and 60 respectively. Use frequency density for your heights.

Answer: 2, 4, 3

Question 7: A teacher reports that a histogram with unequal class widths looks skewed. The data shows class intervals 0-5, 5-15, 15-25 with frequencies 5, 20, 15. The heights are plotted as 1, 2, 3. Which mistake might they have made?

Answer: They plotted heights as frequencies instead of frequency densities.

Question 8: Given class intervals 0-8, 8-16, 16-24, with frequencies 16, 32, 48, calculate the total area of the histogram if plotted with correct heights based on frequency density.

Answer: 128 units squared

Question 9: In a real-world survey, the data shows that the class interval 0-10 has a frequency of 25, and 10-30 has a frequency of 50. Class widths are 10 and 20, respectively. Which class interval has the higher frequency density and why?

Answer: 0-10, because its frequency density is 2.5 compared to 2.5 for 10-30, so they are equal, but if considering the densities, they are the same.

Question 10: Identify a common misconception students might have when constructing histograms with unequal class widths and suggest how to correct it.

Answer: Students often plot bars with heights equal to frequency rather than frequency density. To correct this, they should divide frequency by class width to get the height.

Question 11: Given the data: classes 5-10, 10-20, 20-30 with frequencies 12, 24, 36, calculate the approximate total area of the histogram.

Answer: Total area ≈ 60 units squared