Practice Maths

L57 — Statistical Problem Solving

Key Terms

statistical inquiry cycle
The structured process for a data investigation: pose a question → collect/source data → display the data → calculate measures → interpret → conclude.
conclusion
A statement that directly answers the original statistical question, supported by specific data values. Never claim more than the data shows.
misleading statistics
Data presented in a way that creates a false impression — e.g., manipulated graph scales, missing context, or reporting only selected figures.
interpretation
Explaining what the calculated measures and displays mean in the real-world context of the investigation.
Hot Tip: Your conclusion MUST answer the original statistical question and stay within what the data supports. Never claim more than your data shows — phrases like “this suggests…” or “the data indicates…” are more accurate than “this proves…”

Worked Example

Context: These are the heights (cm) of 10 Year 7 students: {151, 158, 163, 147, 155, 162, 149, 170, 154, 161}.

Step 1 — Choose a display.
A stem-and-leaf plot is appropriate: it shows individual values and the distribution shape for this small continuous dataset.

14 | 7 9
15 | 1 4 5 8
16 | 1 2 3
17 | 0
Key: 14|7 = 147 cm

Step 2 — Calculate measures.
Sorted: 147, 149, 151, 154, 155, 158, 161, 162, 163, 170
Mean = 1570 ÷ 10 = 157.0 cm
Median = (155 + 158) ÷ 2 = 156.5 cm
Mode = none
Range = 170 − 147 = 23 cm

Step 3 — Write conclusions.
1. "The average height of students in this group is about 157 cm."
2. "Most students are clustered in the 150–163 cm range; one student at 170 cm is notably taller than the rest."

The Statistical Inquiry Cycle

Professional statisticians, scientists, and researchers all follow a structured process when investigating a question with data. In Year 7, you learn this same process — called the statistical inquiry cycle. It has five stages, and going through all of them is what separates a complete statistical investigation from simply "making a graph and finding the average."

  1. Pose a clear statistical question.
  2. Collect or source data.
  3. Organise and display the data.
  4. Calculate measures of centre and spread.
  5. Interpret the results and write a conclusion.

Stage 1 — Pose a Clear Statistical Question

A statistical question is one that can be answered with data and that expects variation in the answers. "How tall am I?" is not a statistical question — it has one answer. "How tall are the students in Year 7?" is a statistical question — you need to collect heights from many students and expect different values. Your question should be specific: "Do Year 7 students get more sleep on weekdays or weekends?" is much better than "Do students sleep a lot?"

Stages 2 and 3 — Collecting, Organising, and Displaying

Once you have a question, decide how to collect your data. Will you conduct a survey? Run an experiment? Use secondary data from a reliable source? Think about how to avoid bias — will your sample be representative of the group you're studying?

Once data is collected, organise it into a display that suits the data type and the question. Label everything clearly: title, axis labels, key (if needed). The display should make the answer to your question easier to see — if it doesn't, reconsider your choice of display.

Stages 4 and 5 — Calculating and Communicating

Calculate the mean, median, mode, and range. Choose the measure of centre that best represents the data (median if there are outliers, mean if the data is roughly symmetric). Then write your conclusions. A strong conclusion answers the original question, references specific numbers, and acknowledges the limitations of the investigation. Don't claim more than your data supports — use language like "the data suggests..." or "this sample indicates..." rather than "this proves..."

The conclusion is worth the most marks: In extended data questions, the conclusion is where most of the higher-order marks are awarded. A weak conclusion says "Group A had a bigger mean." A strong conclusion says "Group A had a mean score of 72 compared to Group B's mean of 64, which suggests Group A performed better overall on this test. However, the sample was only 20 students, so these results may not reflect the wider Year 7 population." Practice writing conclusions that include a number, a comparison, and a limitation every time.

  1. Full Statistical Analysis — Race Times Problem Solving

    The following are the times (in seconds) recorded for 10 runners in a 100 m sprint race:

    {13.8, 12.4, 14.7, 11.9, 15.2, 13.1, 12.8, 14.3, 13.5, 12.6}

    1. Choose an appropriate display type for this data and explain your choice.
    2. Calculate the mean, median, mode, and range. Show all working.
    3. Write three conclusions about the runners' performance based on your analysis. Use phrases such as "this suggests..." or "on average..."

  2. Comparing Two Groups — Quiz Scores Problem Solving

    Two classes sat the same quiz out of 40 marks.

    Class A: {28, 35, 22, 38, 31, 25, 37, 30, 27, 33}

    Class B: {18, 40, 15, 39, 20, 38, 17, 36, 22, 35}

    1. Calculate the mean, median, and range for each class.
    2. Which class performed better overall? Use two different measures to justify your answer.
    3. Which class had more consistent results? Explain using a specific statistic.
    4. Write a paragraph comparing the two classes' performances.

  3. Misleading Statistics — Bar Chart Analysis Problem Solving

    A company produces a bar chart showing their sales figures (in thousands of dollars) for four quarters:

    QuarterQ1Q2Q3Q4
    Sales ($'000)82858789

    In the chart, the y-axis starts at 80 (not 0), making Q4 appear to have sales roughly four times higher than Q1 visually.

    1. By what percentage did sales actually increase from Q1 to Q4?
    2. Identify the misleading feature of this chart and explain how it distorts the data.
    3. Describe what a more honest representation of this data would look like.
    4. Why might a company choose to display data in a misleading way?

  4. Dice Experiment — Frequency Analysis Problem Solving

    A fair six-sided die was rolled 40 times. The results were:

    {3, 1, 4, 6, 2, 5, 3, 4, 1, 6, 2, 3, 5, 4, 6, 1, 3, 2, 4, 5, 6, 3, 1, 4, 2, 5, 6, 3, 1, 4, 2, 6, 5, 3, 4, 1, 2, 6, 5, 4}

    1. Create a frequency table showing how many times each number (1–6) was rolled.
    2. If the die is fair, what frequency would you expect for each number in 40 rolls?
    3. Calculate the mean of the 40 rolled values. What would the theoretical mean be if the die is perfectly fair?
    4. Based on your frequency table, does the die appear to be fair? Explain your answer.

  5. Design a Statistical Inquiry — Bedtime Study Problem Solving

    You want to investigate: "What time do Year 7 students go to bed on school nights?"

    1. Write a precise statistical question for this inquiry.
    2. Describe your data collection method: who would you survey, how many people, and how would you ensure the sample is representative?
    3. What display type would you use for the data? Explain why.
    4. Which measures of centre and spread would you calculate, and what would each tell you about bedtimes?

  6. Interpret a Data Table Problem Solving

    The table below shows the number of hours five students spent studying each day for a week.

    StudentMonTueWedThuFri
    Aisha23244
    Ben11512
    Chloe33333
    Dan04042
    Ela22227
    1. Calculate the mean study hours per day for each student.
    2. Calculate the range of daily study hours for each student.
    3. Which student is the most consistent? How do you know?
    4. Ben says he studies "about 2 hours a day on average." Is this accurate? Use evidence from the data.

  7. True or False? Statistics in the Media Problem Solving

    State whether each statistical claim is True, False, or Possibly misleading. Give a reason for each.

    1. "The average Australian earns $120,000 per year." (The median income is $65,000.)
    2. "Our product has a 90% satisfaction rate." (This is based on a survey of only 10 customers.)
    3. "Crime has dropped by 50%." (Last year 2 crimes occurred; this year 1 crime occurred.)
    4. "On average, our patients recover in 3 days." (Some recover in 1 day; some take 2 weeks.)

  8. Spot the Error — Flawed Conclusion Problem Solving

    A student collected data on the heights (cm) of 8 plants and calculated the following:

    Plant12345678
    Height (cm)1215111413161260

    Student's conclusion: "The mean height of plants is 19.1 cm, so a typical plant in this experiment is about 19 cm tall."

    1. Calculate the mean. Is it approximately 19.1 cm?
    2. Identify what is wrong with the student's conclusion.
    3. Which measure would better describe the typical plant height? Calculate and use it to write a more accurate conclusion.

  9. Which Display is Best? Problem Solving

    For each type of data below, choose the most appropriate display (stem-and-leaf plot, dot plot, bar chart, or line graph). Explain your choice.

    DataBest DisplayReason
    Monthly rainfall over 12 months??
    Test scores for 25 students (values 0–100)??
    Favourite sports of 30 students (categories)??
    Number of books read by 15 students (small whole numbers)??

  10. Full Investigation — Screen Time Problem Solving

    A health researcher surveyed 12 teenagers about their daily screen time (hours). The results were:

    {3, 5, 4, 7, 2, 6, 5, 8, 4, 5, 3, 9}

    1. Calculate the mean, median, mode, and range.
    2. The researcher says: "Most teenagers spend about 5 hours on screens each day." Which measure supports this claim? Is it accurate?
    3. The health guidelines recommend no more than 2 hours of recreational screen time per day. How many of these teenagers exceed this recommendation? Express your answer as a fraction and a percentage.
    4. Write a two-paragraph report: the first paragraph describes the data using statistics; the second evaluates what it tells us about teenagers' screen time habits.
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