Mathematics 9709/51 — October/November 2023

Walkthrough

The interquartile range measures the spread of the middle 50% of the data. To find it from a cumulative frequency graph, we first identify the cumulative frequencies that correspond to the lower and upper quartiles.

Step 1: Find the quartile cumulative frequencies.
For $n = 120$ data points:

• The lower quartile (LQ) is the value below which 25% of the data falls, so we look at cumulative frequency $\frac{1}{4} \times 120 = 30$.
• The upper quartile (UQ) is the value below which 75% of the data falls, so we look at cumulative frequency $\frac{3}{4} \times 120 = 90$.

Step 2: Read the time values from the graph.

• At cumulative frequency 30, draw a horizontal line to the curve and then drop a vertical line to the x-axis. This gives $\text{LQ} \approx 23.7$ seconds.
• At cumulative frequency 90, draw a horizontal line to the curve and then drop a vertical line to the x-axis. This gives $\text{UQ} \approx 31$ seconds.

Step 3: Calculate the IQR.
Subtract the lower quartile from the upper quartile: $\text{IQR} = 31 - 23.7 = 7.3$ seconds.

Key Takeaways

• For a cumulative frequency graph with total frequency $n$, the lower quartile is read at cumulative frequency $\frac{n}{4}$ and the upper quartile at $\frac{3n}{4}$.
• The interquartile range is always $\text{UQ} - \text{LQ}$, representing the spread of the middle half of the data.
• Reading values from a cumulative frequency graph requires drawing horizontal lines from the y-axis to the curve and vertical lines down to the x-axis.

Common Mistakes

• Using the wrong cumulative frequency for the quartiles (e.g., using 60 for the lower quartile instead of 30).
• Forgetting to show evidence of reading from the graph — the mark scheme requires visible graph use.
• Subtracting in the wrong order (LQ from UQ instead of UQ from LQ), giving a negative result.

Things to Be Careful About

• Reading values from the graph is inherently approximate. The mark scheme allows $7.0 \leqslant \text{IQR} \leqslant 7.5$, with UQ between 30.5 and 31.25 and LQ between 23.25 and 24. Ensure your readings are consistent with the graph's scale.
• The minor grid lines on the x-axis are every 2 units and on the y-axis every 4 units, so read values carefully to avoid misreading.
• Always show the working: write down the quartile cumulative frequencies and the subtraction step, even if the final answer is within the acceptable range.

Walkthrough

Step 1: Understand the percentage relationship.
The question states that 35% of children took longer than $T$ seconds. This means the remaining 65% took $T$ seconds or less. Since a cumulative frequency graph shows the number of values less than or equal to a given value, we need the cumulative frequency for 65% of the total.

Step 2: Calculate the target cumulative frequency.

This means 78 children took 28.5 seconds or less.

Step 3: Read the value from the graph.
Locate cumulative frequency 78 on the y-axis. Draw a horizontal line to the right until it meets the curve. From that intersection point, draw a vertical line down to the x-axis. The x-axis value at this point is approximately $T \approx 28.5$ seconds.

Key Takeaways

• When a percentage of data is above a value, subtract from 100% to find the percentage below or equal to that value.
• Cumulative frequency graphs directly give the number of observations below a given value, making them ideal for finding percentiles and estimating values from percentages.

Common Mistakes

• Using 35% instead of 65% to find the cumulative frequency, reading the graph at 42 instead of 78.
• Misreading the cumulative frequency on the y-axis — the minor grid lines are every 4 units, so 78 is between 76 and 80.

Things to Be Careful About

• The mark scheme accepts $28 < T < 29$, so readings between 28.1 and 28.9 are all valid. Be precise when reading from the graph.
• Ensure you interpret "longer than $T$" correctly: this is the upper tail, so the cumulative frequency is $100\% - 35\% = 65\%$, not 35%.