Evaluating Data Displays — Solutions

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1. Match display to context

   1. Daily maximum temperature over 30 days: Line graph (continuous data over time)
   2. Favourite colours of 25 students: Column graph (categorical data, comparing counts)
   3. Proportion of weekly time across activities: Pie chart (parts of a whole)
   4. Heights of 20 students showing distribution shape: Stem-and-leaf plot (preserves values, shows shape)
   5. Quiz scores out of 10, every individual score shown: Dot plot (small dataset, discrete values 0–10)
   6. Goals scored per match over 15 matches: Dot plot (small discrete dataset)
   7. Exam results grouped into intervals: Histogram (continuous data grouped into class intervals)
   8. Monthly rainfall totals over one year: Column graph (discrete monthly totals) or line graph (if showing trend over time)

2. Read and interpret data displays

   1. Students who read more than 2 books (3, 4, or 5 books): 4 students (scores of 3, 3, 4, 4, 5)
   2. More students own dogs than cats: 14 − 11 = 3 more students
   3. Students who ride a bike: 10% × 200 = 20 students
   4. Values: 23, 24, 27, 31, 35, 35, 38, 40, 42, 46, 53. Median (6th value): 35
   5. Largest increase in savings: April to May: increased by $20 (from $90 to $110)
   6. Students who scored 70 or above: 12 + 6 = 18 students
   7. Total sales: $12 400 + $13 100 + $11 900 = $37 400
   8. Flathead caught: 30% × 80 = 24 flathead

3. Identify misleading features

   1. Truncated y-axis starting at $950 makes the difference between $980 and $1020 (only $40) appear very large visually.
   2. 3D perspective distorts the size of pie slices — front slices appear larger than equally-sized back slices due to angle and depth.
   3. Unequal time intervals on the x-axis but equal spacing between them makes the rate of change look consistent when it is not.
   4. Different bar widths mean bars cover different areas — a wide bar looks like it represents more data even if the height (frequency density) is the same.
   5. Y-axis from 60 to 70 makes the difference between 62% and 68% look much larger than 6 percentage points out of 100.
   6. Percentages totalling 110% is mathematically impossible for a pie chart; each sector is inflated to make the proportions appear more extreme.
   7. Omitting the y-axis scale makes it impossible to determine actual values, allowing any impression to be created by bar heights alone.
   8. An unlabelled axis break makes it appear the y-axis starts at 0 when it does not, exaggerating differences between values.

4. Compare two displays

   1. Display B (line graph) is more appropriate. Sunshine hours over 30 consecutive days shows change over time, which a line graph represents clearly. A dot plot does not show time order.
   2. Display B (pie chart) is more appropriate. A pie chart shows proportions of a whole budget clearly. A column graph is better for comparing independent categories, not parts of a total.
   3. Display B (stem-and-leaf plot) is more appropriate. With 30 scores ranging from 42 to 98, a dot plot would be very cluttered. A stem-and-leaf plot groups by tens, making the distribution easy to read.
   4. Display B (back-to-back stem-and-leaf plot) is more appropriate. It allows direct comparison of both groups on the same scale and preserves every data value.
   5. Display B (line graph) is more appropriate. The data shows change across 24 time periods. A pie chart would be meaningless here as hourly counts are not parts of a meaningful whole.
   6. Display B (column graph) is more appropriate. Letter grades are discrete categories. A histogram is for continuous data grouped into class intervals and would be misleading for labelled categories.

5. Fix misleading displays

   1. Start the y-axis at 0 so that bars representing 97 and 99 are seen in context and the tiny difference (2 out of 100) is not visually exaggerated.
   2. Redraw as a flat (2D) bar chart. This removes depth distortion caused by 3D perspective and makes all bar heights directly comparable.
   3. Recalculate each segment so percentages sum to exactly 100%. Each slice should represent its true proportion of the whole dataset.
   4. Space the x-axis intervals proportionally to the actual time gaps (e.g., 1 year gap = 1 unit, 5 year gap = 5 units) so the gradient of the line reflects the true rate of change.
   5. Replace pictographs of different-sized people with equal-sized symbols or use a standard column graph where height or count alone represents the value.
   6. Use the same scale on both y-axes (or a single y-axis), so that equal changes in improvement are represented by equal visual distances on the graph.

6. Critique statistical claims

   1. The claim is not accurate. Sales increased from 4600 to 4900 units, a rise of 300 units. That is a 6.5% increase, not “double.” The bar graph’s truncated y-axis (starting at 4500) makes the second bar appear approximately twice the height of the first, which visually suggests doubling but is mathematically false.
   2. Problem 1: The sample is too small (n = 40) to represent an entire suburb or city population reliably. Problem 2: Surveying only shoppers at a shopping centre on a Tuesday morning introduces selection bias — this group is not representative of all residents (e.g. it excludes people who are working, housebound, or do not shop at that centre).
   3. The graph appears to show crime rising steeply because the y-axis runs from 980 to 1020. In reality, crime increased from 988 to 1008 incidents — a rise of 20 incidents over six years, roughly a 2% increase. The compressed y-axis makes a gradual, modest rise appear dramatic. The display should use a y-axis starting at 0 (or at minimum clearly label and justify the truncation) and the headline should be reworded to accurately reflect the small absolute and percentage change.

7. True or False about data displays

   1. False — a histogram is used for continuous (or grouped numerical) data, not categorical data.
   2. False — a line graph shows change over time; a pie chart shows parts of a whole.
   3. True.
   4. False — a y-axis starting at 0 can still be misleading if, for example, the scale is compressed, intervals are unequal, or data points are selectively highlighted.
   5. False — a dot plot becomes cluttered with large datasets; stem-and-leaf plots or histograms are more appropriate for larger datasets.
   6. True.
   7. False — a pie chart shows parts of a whole better; a column graph compares separate categories.
   8. True.

8. Read a back-to-back stem-and-leaf plot

   1. Class A: 3 + 4 + 3 + 2 + 1 = 13 students; Class B: 3 + 4 + 4 + 3 + 2 = 16 students
   2. Class A highest: 92; Class B highest: 94
   3. Class A has 13 values; median is the 7th value. Listing in order: 55, 57, 59, 62, 63, 66, 68, 71, 75, 79, 84, 88, 92. Median = 68
   4. Class B has 16 values; median is average of 8th and 9th. Listing in order: 52, 54, 58, 61, 63, 65, 69, 70, 73, 76, 77, 82, 85, 88, 91, 94. Median = (70 + 73) ÷ 2 = 71.5
   5. Class B performed better overall. Its median (71.5) is higher than Class A’s median (68), and Class B has more scores in the 70s and 80s, indicating a higher concentration of scores above average.
   6. Class B scores of 70 or more: 70, 73, 76, 77, 82, 85, 88, 91, 94 = 9 students

9. Choose and justify a display

   1. Line graph — it shows how a continuous quantity (water level) changes over ordered time periods.
   2. Dot plot or stem-and-leaf plot — the dataset is small (30 students) and discrete (0–10), so individual values can be displayed; a stem-and-leaf plot also shows the shape of the distribution.
   3. Pie chart — the three categories are parts of a whole (they sum to 100%), which a pie chart represents most clearly.
   4. Back-to-back stem-and-leaf plot — it allows direct comparison of two datasets of the same size while preserving all original values.
   5. Column graph — the data is categorical (states) with a numerical count per category; a column graph allows easy comparison across all states.

10. Create and critique data displays

    1. (i) Stem-and-leaf plot:
       5 | 8
       6 | 2 4 5 7
       7 | 0 2 4 8
       8 | 1 5
       9 | 1
       (ii) The distribution is slightly skewed right (most scores are in the 60s and 70s, with fewer high scores).
       (iii) Starting the y-axis at 55 would make small differences in scores look enormous. For example, a score of 91 and 58 would appear 6× further apart than they really are proportionally. This is misleading because it exaggerates variation.
    2. (i) Increase: 4.3 − 3.8 = 0.5 hours
       (ii) Percentage increase: (0.5 ÷ 3.8) × 100 ≈ 13.2%
       (iii) “Skyrocketed” is not justified. Usage increased by only 0.5 hours (13%) over 4 years — a gradual, modest rise. The truncated y-axis (3.5 to 4.5) makes this small increase appear dramatic.
       (iv) The y-axis should start at 0 so that the bars or line segments are proportional to the actual values. This would make the 0.5-hour increase look appropriately small relative to a baseline of 0.