Mastering Cumulative Frequency Graphs and Box Plots for your Maths Exam

What the examiner is testing

The examiner is testing your ability to construct an accurate cumulative frequency graph from a frequency table and use it to estimate medians, quartiles, and interquartile ranges. Marks are awarded for correctly plotting points, drawing a smooth curve, and accurately reading values from your graph. You are also assessed on your ability to draw and interpret box plots, comparing distributions.

The method

1. Construct the cumulative frequency table: Add a new column to your frequency table. For each row, the cumulative frequency is the sum of the frequency for that row and all frequencies above it. The last cumulative frequency should equal the total frequency.
2. Determine plotting points: Plot the upper class boundary of each interval against its cumulative frequency. For example, if an interval is \(10 < x \le 20\) with a cumulative frequency of 35, plot the point \((20, 35)\). Always include a starting point at the lower bound of the first interval with a cumulative frequency of 0 (e.g., if the first interval is \(0 < x \le 10\), plot \((0, 0)\)).
3. Draw the cumulative frequency graph: Plot all your points accurately. Draw a smooth, S-shaped curve through these points. Do not use a ruler to connect points with straight lines. The curve should start at the origin (or the lower bound of the first interval) and end at the total cumulative frequency.
4. Estimate the median: Locate the value on the cumulative frequency axis that corresponds to half of the total frequency. Draw a horizontal line from this point to your curve, then a vertical line down to the x-axis. Read the value on the x-axis; this is your estimated median.
5. Estimate the lower quartile (LQ): Locate the value on the cumulative frequency axis that corresponds to one-quarter of the total frequency. Draw a horizontal line to the curve, then a vertical line down to the x-axis. Read this value.
6. Estimate the upper quartile (UQ): Locate the value on the cumulative frequency axis that corresponds to three-quarters of the total frequency. Draw a horizontal line to the curve, then a vertical line down to the x-axis. Read this value.
7. Calculate the interquartile range (IQR): Subtract the lower quartile from the upper quartile: \(IQR = UQ - LQ\).
8. Draw a box plot:
   • Find the minimum and maximum values from the original data (or the lowest and highest values given in the question).
   • Draw a number line covering the range of your data.
   • Mark the minimum value, lower quartile, median, upper quartile, and maximum value with vertical lines above the number line.
   • Draw a box from the lower quartile to the upper quartile.
   • Draw a line inside the box at the median.
   • Draw "whiskers" from the box to the minimum and maximum values.

Worked example

The table shows the heights of 80 students. Draw a cumulative frequency graph and use it to estimate the median height.

1. Construct the cumulative frequency table:

2. Determine plotting points:
   \((140, 0)\) (starting point)
   \((150, 8)\)
   \((160, 32)\)
   \((170, 68)\)
   \((180, 80)\)

3. Draw the cumulative frequency graph: (Imagine a graph with height on the x-axis from 140 to 180, and cumulative frequency on the y-axis from 0 to 80. Points are plotted as above and connected with a smooth S-shaped curve.)

4. Estimate the median:
   Total frequency \( = 80 \).
   Median position \( = \frac{80}{2} = 40^{th} \) value.
   Locate 40 on the cumulative frequency axis.
   Draw a horizontal line from 40 to the curve.
   Draw a vertical line down to the height axis.
   Read the value.

   Estimated Median \( \approx 162.5 \) cm.

   Sanity check: The median should fall within the interval \(160 < h \le 170\), as this interval contains the 33rd to 68th values. \(162.5\) cm is within this range.

Worked example: a harder one

The cumulative frequency graph shows the times, in minutes, that 100 people waited for a bus.

(Graph description: x-axis "Time (minutes)" from 0 to 30. y-axis "Cumulative Frequency" from 0 to 100. A smooth S-shaped curve passes through (0,0), (5,10), (10,30), (15,65), (20,85), (25,95), (30,100).)

Use the graph to estimate the interquartile range and the number of people who waited longer than 22 minutes.

1. Estimate the lower quartile (LQ):
   Total frequency \( = 100 \).
   LQ position \( = \frac{1}{4} \times 100 = 25^{th} \) value.
   From the graph, locate 25 on the cumulative frequency axis. Draw horizontally to the curve, then vertically down.
   LQ \( \approx 9 \) minutes.

2. Estimate the upper quartile (UQ):
   UQ position \( = \frac{3}{4} \times 100 = 75^{th} \) value.
   From the graph, locate 75 on the cumulative frequency axis. Draw horizontally to the curve, then vertically down.
   UQ \( \approx 17.5 \) minutes.

3. Calculate the interquartile range (IQR):
   IQR \( = UQ - LQ \)
   IQR \( = 17.5 - 9 \)
   IQR \( = 8.5 \) minutes.

4. Estimate the number of people who waited longer than 22 minutes:
   Obvious first move fails: You might be tempted to find 22 minutes on the x-axis and read the cumulative frequency, then state that as the answer. This gives the number of people who waited up to 22 minutes, not longer than.

   Correct approach:
   Locate 22 minutes on the time axis.
   Draw a vertical line up from 22 to the curve.
   Draw a horizontal line from the curve to the cumulative frequency axis.
   Read the value: This is the number of people who waited up to 22 minutes. Let's say this value is \( \approx 90 \).
   The total number of people is 100.
   Number of people who waited longer than 22 minutes \( = \text{Total people} - \text{People who waited up to 22 minutes} \)
   Number of people \( = 100 - 90 = 10 \) people.

Practice

1. The table shows the masses of 50 apples.
   | Mass (\(m\) g) | Frequency |
   | :-------------- | :-------- |
   | \(100 < m \le 120\) | 5 |
   | \(120 < m \le 140\) | 15 |
   | \(140 < m \le 160\) | 22 |
   | \(160 < m \le 180\) | 8 |
   Draw a cumulative frequency graph for this data. Use your graph to estimate the median mass.

2. Using your graph from question 1, estimate the interquartile range.

3. The cumulative frequency graph shows the distances, in km, that 60 cyclists rode.
   (Graph description: x-axis "Distance (km)" from 0 to 100. y-axis "Cumulative Frequency" from 0 to 60. A smooth S-shaped curve passes through (0,0), (20,8), (40,25), (60,45), (80,55), (100,60).)
   Estimate the number of cyclists who rode between 30 km and 70 km.

4. The box plot shows the distribution of scores for Class A in a test.
   (Box plot description: A number line from 0 to 100. Whiskers from 20 to 95. Box from 40 to 80. Median line at 65.)
   Class B had a median score of 70, a lower quartile of 50, an upper quartile of 85, a minimum score of 30, and a maximum score of 90.
   Compare the distribution of scores for Class A and Class B.

Answers

1. Cumulative Frequency Table:
   | Mass (\(m\) g) | Frequency | Cumulative Frequency |
   | :-------------- | :-------- | :------------------- |
   | \(100 < m \le 120\) | 5 | 5 |
   | \(120 < m \le 140\) | 15 | 20 |
   | \(140 < m \le 160\) | 22 | 42 |
   | \(160 < m \le 180\) | 8 | 50 |
   Plotting points: \((100,0), (120,5), (140,20), (160,42), (180,50)\).
   (Graph would be drawn with these points and a smooth curve.)
   Median: Total frequency \( = 50 \). Median position \( = \frac{50}{2} = 25^{th} \) value.
   From the graph, reading across from 25 on the cumulative frequency axis and down to the mass axis:
   Estimated Median \( \approx 143 \) g. (Acceptable range 142-144g)

2. Lower Quartile (LQ): Position \( = \frac{1}{4} \times 50 = 12.5^{th} \) value.
   From the graph, LQ \( \approx 132 \) g. (Acceptable range 131-133g)
   Upper Quartile (UQ): Position \( = \frac{3}{4} \times 50 = 37.5^{th} \) value.
   From the graph, UQ \( \approx 156 \) g. (Acceptable range 155-157g)
   Interquartile Range (IQR): \( = UQ - LQ = 156 - 132 = 24 \) g. (Acceptable range 22-26g)

3. Number of cyclists who rode between 30 km and 70 km:
   At 30 km, cumulative frequency \( \approx 16 \).
   At 70 km, cumulative frequency \( \approx 50 \).
   Number of cyclists \( = 50 - 16 = 34 \). (Acceptable range 32-36)

4. Comparison:

   • Median: Class B has a higher median score (70) than Class A (65), indicating that on average, Class B performed better.
   • Interquartile Range (IQR):
     Class A IQR \( = 80 - 40 = 40 \).
     Class B IQR \( = 85 - 50 = 35 \).
     Class B has a smaller IQR (35) than Class A (40), meaning the scores in Class B are less spread out or more consistent than in Class A.
   • Range:
     Class A Range \( = 95 - 20 = 75 \).
     Class B Range \( = 90 - 30 = 60 \).
     Class B has a smaller overall range (60) compared to Class A (75), also suggesting less variability in Class B scores.

The three mistakes that lose marks

1. Connecting points with straight lines on a cumulative frequency graph: This implies that the data increases linearly between the upper class boundaries, which is incorrect. The curve should be smooth.
   • Wrong answer produced: A jagged, angular graph instead of a smooth S-curve. This makes estimations of quartiles less accurate.
2. Plotting cumulative frequency against the mid-point or lower class boundary: Cumulative frequency represents "less than or equal to" the upper class boundary.
   • Wrong answer produced: The entire graph is shifted to the left or right, leading to incorrect quartile estimations. For example, plotting \((145, 8)\) instead of \((150, 8)\) for the first interval.
3. Misinterpreting "more than" or "less than" on the graph: When asked for the number of values above a certain point, students often read the cumulative frequency up to that point.
   • Wrong answer produced: If 100 people waited, and 80 waited up to 15 minutes, stating 80 people waited more than 15 minutes instead of \(100 - 80 = 20\) people.

30-second recap

Cumulative frequency graphs show the running total of frequencies, plotted at upper class boundaries, and are used to estimate medians and quartiles. Box plots summarise data using five key values: minimum, lower quartile, median, upper quartile, and maximum, allowing for easy comparison of distributions. Always draw a smooth curve for cumulative frequency and ensure correct interpretation for "more than" questions.