Practice Maths

Evaluating Data Displays

Key Ideas

Key Terms

column graph
used for comparing categories. Each bar represents the count or value for one category.
histogram
shows the frequency of continuous data grouped into equal intervals. Bars touch (no gaps).
line graph
shows change over time. Points are connected by lines to reveal trends.
dot plot
displays small datasets with each data value shown as a dot above a number line.
stem-and-leaf plot
shows the shape of the distribution while preserving original values. Back-to-back version compares two datasets.
pie chart
shows proportions of a whole as sectors of a circle. Best for part-of-a-whole comparisons.
misleading display
a graph that distorts the truth through a truncated y-axis, inconsistent scale intervals, or 3D effects that exaggerate differences.
Hot Tip A y-axis that does not start at 0 can make small differences look enormous. Always check the scale before drawing conclusions from a graph.

Worked Example

Question: A bar graph shows sales of $102 000 and $104 000 for two companies, but the y-axis starts at $100 000. What misleading impression does this create?

Answer: Because the y-axis starts at $100 000 instead of $0, the bar for $104 000 appears to be roughly twice the height of the bar for $102 000. In reality, Company B’s sales are only about 2% higher. The truncated axis exaggerates the difference.

Why Graphs Can Lie

Data displays — graphs, charts, infographics — are supposed to show the truth about data clearly and honestly. But graphs can be constructed in ways that make differences look bigger or smaller than they really are. This is sometimes done deliberately (to persuade people) and sometimes by mistake. As a data-literate person, your job is to look critically at any graph before accepting its conclusion.

The three most common tricks to watch for are: truncated axes, inconsistent scales, and cherry-picked data. Once you know what to look for, you will spot misleading graphs everywhere — in news articles, social media, and advertising.

Truncated Axes

A truncated axis is one that does not start at zero. This is the most common way graphs mislead people. When a bar chart's vertical axis starts at, say, 90 instead of 0, a bar reaching 92 and a bar reaching 95 look dramatically different — but the actual difference is only 3 units.

Example: Imagine two companies' quarterly sales are $98 million and $102 million. If the y-axis starts at $95 million, the second bar looks about five times taller than the first, making the difference seem massive. If the axis starts at 0, both bars are nearly the same height and the $4 million difference is put into honest perspective.

Note: truncated axes are not always wrong — for line graphs showing small changes over time, starting above zero is sometimes reasonable. But you should always check where the axis starts and mentally rescale the differences in your head.

Inconsistent or Misleading Scales

An inconsistent scale occurs when the intervals on an axis are not evenly spaced — for instance, the y-axis might jump from 10 to 20 to 30 and then suddenly to 100. This distorts how the data looks, making some changes appear gradual and others sudden.

Another trick is using 3D charts, which are almost always misleading because the perspective makes front bars appear larger than back bars of the same height. Similarly, pictographs can mislead when a figure is scaled in two dimensions (both height and width), making an object that is "twice as tall" appear four times as large in area.

The rule is: every axis should have a consistent, evenly spaced scale, clearly labelled with units, starting from a value the viewer can interpret fairly.

Cherry-Picked Data

Cherry-picking means selecting only the data that supports your argument and leaving out data that contradicts it. For example, showing a graph of temperature data only from particularly warm years to "prove" a trend, while ignoring cooler years that don't fit the story.

When evaluating a data display, always ask: Is this the full data set, or has someone only selected a convenient portion? How was the time period chosen? Are there other data points that might tell a different story?

A related issue is missing context: "Sales increased by 50%!" sounds impressive — but if it went from 2 items to 3 items, that is a very different claim. Always look for absolute numbers, not just percentages, and ask what the baseline was.

How to Evaluate a Data Display Properly

Use this checklist every time you see a graph:

  • Does the y-axis start at zero? If not, is that justified?
  • Are the axis intervals evenly spaced and clearly labelled with units?
  • Does the title accurately describe what the graph shows?
  • Is the full relevant data included, or has it been cropped in time or category?
  • Is a 3D effect used that could distort the visual impression?
  • Is the source of the data mentioned? Is it reliable?
  • Does the conclusion drawn actually follow from what the graph shows?
Key tip: Whenever a graph makes you feel a strong emotional reaction — "Wow, that's a huge increase!" or "Look how much better that product is!" — that is exactly when you should slow down and look more carefully. The most misleading graphs are the ones designed to trigger a quick, unthinking response. Train yourself to always check the axis scale before drawing any conclusion.

Mastery Practice

  1. Match each data type or context to the most appropriate display. Choose from: column graph, histogram, line graph, dot plot, stem-and-leaf plot, pie chart. Fluency

    1. The daily maximum temperature in Brisbane over 30 days (continuous, time order).
    2. The favourite colours of 25 students (categorical, comparing counts).
    3. The proportion of a student’s weekly time spent on homework, sleep, sport, and leisure.
    4. The heights (in cm) of 20 students, to show the overall shape of the data.
    5. The scores out of 10 on a quiz taken by a class of 28 students, showing every individual score.
    6. The number of goals scored per match by a soccer team over a 15-match season (small dataset, discrete).
    7. The distribution of Year 8 students’ exam results (grouped into intervals 50–59, 60–69, etc.).
    8. The monthly rainfall totals in Cairns over one year (change over time).

  2. Use the described data displays to answer the questions. Fluency

    1. A dot plot shows the number of books read by 10 students last month: 1, 1, 2, 2, 2, 3, 3, 4, 4, 5. How many students read more than 2 books?
    2. A column graph shows pets owned: Dogs = 14, Cats = 11, Fish = 7, Birds = 4, Other = 4. How many more students own dogs than cats?
    3. A pie chart shows that 35% of students walk to school, 40% take the bus, 15% are driven, and 10% ride a bike. If there are 200 students, how many ride a bike?
    4. A stem-and-leaf plot has stems 2, 3, 4, 5 with leaves: 2|3 4 7, 3|1 5 5 8, 4|0 2 6, 5|3. List all values and find the median.
    5. A line graph shows monthly savings: Jan=$50, Feb=$80, Mar=$60, Apr=$90, May=$110. In which month did savings increase the most?
    6. A histogram shows exam scores: 50–59 (4 students), 60–69 (8 students), 70–79 (12 students), 80–89 (6 students). How many students scored 70 or above?
    7. A column graph shows three stores’ sales: Store A = $12 400, Store B = $13 100, Store C = $11 900. What is the total sales across all three stores?
    8. A pie chart shows types of fish caught: Bream 45%, Flathead 30%, Whiting 15%, Other 10%. Of 80 fish caught, how many were flathead?

  3. Identify the misleading feature in each display description. Fluency

    1. A column graph comparing two products’ sales ($980 and $1020) has a y-axis that starts at $950.
    2. A 3D pie chart makes the front slice appear larger than slices of the same percentage at the back.
    3. A line graph shows population growth but the intervals on the x-axis jump from 1990 to 1995 to 1998 to 2000 to 2010 unevenly.
    4. A histogram uses bars of different widths to represent each interval, making wide-bar groups appear more frequent.
    5. An advertisement shows a bar graph where the y-axis goes from 60 to 70 to compare two customer satisfaction scores of 62% and 68%.
    6. A pie chart uses percentages that add to 110% (each segment is slightly exaggerated).
    7. A column graph omits the y-axis scale entirely, making it impossible to read actual values.
    8. A line graph has a break symbol (//) on the y-axis, but this is not clearly labelled, making readers think the y-axis starts at 0.

  4. Two displays of the same data are described. State which is more appropriate and why. Understanding

    1. Data: the number of hours of sunshine per day over 30 days. Display A: a dot plot. Display B: a line graph with days on the x-axis.
    2. Data: the percentage of budget spent on five categories. Display A: a column graph. Display B: a pie chart.
    3. Data: the test scores of 30 students (values from 42 to 98). Display A: a dot plot with 57 possible values shown. Display B: a stem-and-leaf plot.
    4. Data: comparing the heights of boys and girls in Year 8. Display A: two separate stem-and-leaf plots. Display B: a back-to-back stem-and-leaf plot.
    5. Data: the number of cars passing a point each hour over 24 hours. Display A: a pie chart. Display B: a line graph.
    6. Data: the frequency of each letter grade (A, B, C, D, E) in a class assessment. Display A: a histogram. Display B: a column graph.

  5. Describe what changes would make each misleading display fair and accurate. Understanding

    1. A column graph has a y-axis starting at 95 to compare scores of 97 and 99 out of 100.
    2. A 3D bar chart distorts heights due to perspective. How should it be redrawn?
    3. A pie chart has labels showing percentages that sum to 105%.
    4. A line graph has unequal time intervals on the x-axis (1 year, 1 year, 3 years, 5 years) but equal spacing.
    5. A column graph uses pictures of people of different heights to represent numbers, making larger groups look disproportionately bigger.
    6. A graph compares two schools’ NAPLAN improvement but uses different scales on each y-axis.

  6. Critique the following statistical claims and displays. Problem Solving

    1. A cereal advertisement claims: “Sales have doubled in three months!” The bar graph has a y-axis starting at 4500 units, with bars at 4600 (Month 1) and 4900 (Month 3). Is the claim accurate? Explain what the graph shows vs. what the headline claims.
    2. A local council displays a pie chart showing that 75% of residents “support” a new road, but the survey only asked 40 people at a shopping centre on a Tuesday morning. Identify two problems with this conclusion.
    3. A newspaper headline reads: “Crime is rising sharply.” The line graph shows crime incidents from 2018 to 2024, with the y-axis running from 980 to 1020. The actual values are: 2018=988, 2020=995, 2022=1001, 2024=1008. Describe what the graph appears to show, what the data actually shows, and how the display should be improved.

  7. State whether each statement about data displays is True or False. If false, write the correct statement. Understanding

    1. A histogram is used for categorical data grouped into class intervals.
    2. A pie chart is the best choice for showing how a quantity changes over time.
    3. A back-to-back stem-and-leaf plot can be used to compare two datasets.
    4. A y-axis that starts at 0 can never be misleading.
    5. A dot plot works best for large datasets with hundreds of values.
    6. A line graph is appropriate for showing temperature changes over a week.
    7. A column graph shows parts of a whole better than a pie chart.
    8. Unequal intervals on the x-axis of a line graph can make trends appear faster or slower than they really are.

  8. Use the back-to-back stem-and-leaf plot to answer the questions. Understanding

    The plot shows the test scores of Class A (left) and Class B (right).

    Class A  |  Stem  |  Class B
   9 7 5 |   5    |  2 4 8
 8 6 3 2 |   6    |  1 3 5 9
   9 5 1 |   7    |  0 3 6 7
     8 4 |   8    |  2 5 8
       2 |   9    |  1 4
    1. How many students are in each class?
    2. What is the highest score in Class A? In Class B?
    3. What is the median score for Class A?
    4. What is the median score for Class B?
    5. Which class performed better overall? Justify using the median and the spread of scores.
    6. How many students in Class B scored 70 or more?

  9. For each scenario, choose the most appropriate display and explain your choice in one sentence. Understanding

    1. A scientist measures the water level in a dam every month for two years.
    2. A teacher records the number of errors each student made on a spelling test (scores 0–10, class of 30).
    3. A shop owner wants to show that outdoor furniture makes up 55% of total sales, while indoor furniture is 30% and accessories are 15%.
    4. A researcher compares the reaction times (in milliseconds) of 25 teenagers and 25 adults.
    5. A journalist wants to display the number of road accidents per year in each Australian state.

  10. Create and critique data displays. Problem Solving

    1. The scores of 12 students on a maths test are: 58, 62, 64, 65, 67, 70, 72, 74, 78, 81, 85, 91.
      1. Construct a stem-and-leaf plot for this data.
      2. Describe the shape of the distribution (symmetric, skewed left, skewed right).
      3. A student draws a column graph with the y-axis starting at 55. What impression would this give? How is it misleading?
    2. A newspaper claims: “Smartphone use among teenagers has skyrocketed.” The graph shows daily usage (hours): 2019 = 3.8, 2020 = 4.0, 2021 = 4.1, 2022 = 4.2, 2023 = 4.3. The y-axis runs from 3.5 to 4.5.
      1. By how many hours did daily usage actually increase from 2019 to 2023?
      2. What percentage increase does this represent?
      3. Is the word “skyrocketed” justified? Explain using your calculations.
      4. Describe how the y-axis could be changed to give a more accurate visual impression.
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