Mathematics 9709/63 — May/June 2018

Walkthrough

First, we calculate the total number of children recorded, which is the sum of all frequencies: $6 + 12 + 14 + 10 + 8 = 50$. The lower quartile (Q1) is the value below which 25% of the data falls. For 50 data points, the lower quartile position is $\frac{50}{4} = 12.5$.

Next, we build up the cumulative frequency table to see where the 12.5th value lies. The first class (10–14) has 6 children, so cumulative frequency is 6. Adding the second class (15–19) with 12 children gives a cumulative frequency of 18. Since $6 < 12.5 \leq 18$, the lower quartile must lie in the 15–19 class interval.

Key Takeaways

• The lower quartile position for $N$ data points is $N/4$.
• Cumulative frequencies help identify which class interval contains a given percentile.
• For grouped data, we can only identify the class interval, not the exact value, of a quartile.

Common Mistakes

• Using $N/4 + 1$ instead of $N/4$ for the lower quartile position.
• Forgetting that class boundaries for continuous data are at the half-integers (e.g., 14.5, 19.5), so the class 15–19 has upper boundary 19.5.
• Miscounting cumulative frequencies.

Things to Be Careful About

• The class intervals are $10$–$14$, $15$–$19$, $20$–$24$, $25$–$34$, $35$–$59$. These are continuous intervals with boundaries at $9.5, 14.5, 19.5, 24.5, 34.5, 59.5$.
• The question accepts $15$–$19$ or $14.5$–$19.5$ as the answer.
• The unit (kg) is not required for the mark.

Walkthrough

When drawing a histogram for grouped data with unequal class widths, the key principle is that the area of each bar must be proportional to the frequency. This means:

• Bar width = class width
• Bar height = frequency density = frequency / class width
• Area = width × height = class width × (frequency / class width) = frequency

Step 1: Identify the class boundaries. Since the data is continuous (masses recorded to the nearest kg), the boundaries are at the half-integers: 9.5, 14.5, 19.5, 24.5, 34.5, 59.5.

Step 2: Calculate the class widths:

• 10–14: $14.5 - 9.5 = 5$
• 15–19: $19.5 - 14.5 = 5$
• 20–24: $24.5 - 19.5 = 5$
• 25–34: $34.5 - 24.5 = 10$
• 35–59: $59.5 - 34.5 = 25$

Step 3: Calculate the frequency densities (fd = frequency / class width):

• $6 / 5 = 1.2$
• $12 / 5 = 2.4$
• $14 / 5 = 2.8$
• $10 / 10 = 1.0$
• $8 / 25 = 0.32$

Step 4: Draw the histogram on the grid:

• The horizontal axis must be labelled "Mass (kg)" and range from at least 9.5 to 59.5 with a linear scale.
• The vertical axis must be labelled "fd" (or "frequency density") and range from 0 to at least 3, with a linear scale and at least 3 equally spaced values marked.
• Draw five bars with no gaps between them, starting at x = 9.5.
• The first four bars have equal width (proportional to 5), and the last two have widths in ratio 10:25 = 2:5 relative to the first.
• The heights are 1.2, 2.4, 2.8, 1.0, and 0.32 respectively.

Key Takeaways

• For histograms with unequal class widths, always use frequency density on the vertical axis.
• Frequency density = frequency / class width.
• Bar widths must be proportional to class widths, and bars must have no gaps.
• The horizontal axis must start at the lower class boundary (9.5), not at zero, unless a break is indicated.

Common Mistakes

• Using frequency instead of frequency density on the vertical axis (this gives incorrect bar heights for unequal class widths).
• Forgetting that the class 35–59 has width 25, not 24 or 20.
• Drawing bars with gaps between them (histograms for continuous data have no gaps).
• Starting the horizontal axis at 0 without indicating a break, when the data starts at 9.5.
• Not labelling both axes correctly (must include "Mass (kg)" and "fd" or "frequency density").

Things to Be Careful About

• The class widths are 5, 5, 5, 10, 25 — these are not equal, so frequency density must be used.
• The horizontal axis must range from at least 9.5 to 59.5. If it starts from zero, a break in the scale must be indicated.
• The vertical axis must be linear with at least 3 equally spaced values marked.
• At least 3 linearly spaced values must appear on each axis for full marks.