Practice Maths

L55 — Mixed Data Analysis

A complete statistical inquiry follows this cycle:

  1. Pose a question
  2. Collect or obtain data
  3. Display the data
  4. Calculate measures (mean, median, mode, range)
  5. Interpret — what do the measures tell you?
  6. Conclude — answer the original question
Hot Tip: A complete analysis always includes an appropriate display, measures of centre AND spread, and interpretation in context. Missing any of these makes your analysis incomplete.

Worked Example

Dataset: {45, 52, 67, 71, 43, 58, 72, 65, 49, 76, 61, 54}

Sorted: 43, 45, 49, 52, 54, 58, 61, 65, 67, 71, 72, 76

4 | 3 5 9
5 | 2 4 8
6 | 1 5 7
7 | 1 2 6    Key: 4|3 = 43

Mean = 713 ÷ 12 = 59.4  |  Median = (58+61)÷2 = 59.5  |  No mode  |  Range = 76−43 = 33

Conclusions: "The class averaged around 59–60, suggesting mid-range achievement. The range of 33 marks indicates moderate variability."

Key Terms

statistical inquiry cycle
the process of posing a question, collecting data, displaying it, calculating measures, interpreting, and concluding
conclusion
a statement that answers the original question using specific data values; must stay within what the data shows
limitation
a factor that restricts what conclusions can be drawn — e.g. small sample, biased collection, or data from a different context
over-generalise
to draw conclusions that go further than the data supports — e.g. using one class's results to claim something about all students in Australia

Putting It All Together

This lesson applies everything from Year 7 Data. A complete analysis has three stages: organise and display, calculate measures, then interpret and communicate. Skipping any stage makes your analysis incomplete.

Stage 1 — Choose and Create a Display

Ask: what type of data is this, and what am I trying to show?

  • Stem-and-leaf / dot plot — small numerical datasets; shows full distribution
  • Back-to-back stem-and-leaf — comparing two groups
  • Line graph — showing how a quantity changes over time
  • Column / bar graph — comparing categories

Always include a title, labelled axes, and a key if needed.

Stage 2 — Calculate the Measures

For a complete statistical summary:

  • Mean — sum ÷ count
  • Median — middle value in sorted data
  • Mode — most frequent value
  • Range — maximum − minimum

If there is an outlier, note it and consider whether mean or median better represents the data.

Stage 3 — Interpret and Write Conclusions

A good conclusion:

  • Answers the original question using actual numbers from the data
  • Uses language like "the data suggests...", "on average...", "the results indicate..."
  • Does not over-generalise beyond the sample
  • Notes any limitations — small sample? biased collection? outliers?

When comparing two groups, structure your conclusion as: "Group A had a [higher/lower] [measure] of [value] compared to Group B's [measure] of [value], suggesting..."

Common Mistakes to Avoid

  • Forgetting to sort data before finding the median
  • Using the mean when there is a clear outlier
  • Writing "Group A is better" without saying by how much or in what way
  • Drawing a line graph for categorical data
  • Confusing "no mode" with "the mode is 0"
Exam structure: When asked to "analyse the data and draw conclusions": (1) State your display type and why; (2) Calculate mean, median, mode, range with working; (3) Write 2–3 conclusion sentences referencing specific numbers; (4) Identify at least one limitation.

  1. Calculate All Four Measures

    For each dataset, find the mean, median, mode, and range.

    1. Dataset A — Goals scored per game (10 games): {3, 5, 2, 4, 5, 1, 3, 6, 5, 2}
    2. Dataset B — Heights (cm) of 12 students: {148, 152, 165, 147, 158, 152, 171, 149, 162, 152, 155, 161}
    3. Dataset C — Time (min) to complete a puzzle (14 participants): {8, 12, 7, 15, 9, 11, 8, 14, 10, 8, 13, 6, 11, 9}

  2. Choose the Best Display Type

    For each dataset, choose the best display type and explain your choice. (Display types: dot plot, stem-and-leaf plot, line graph, column/bar graph, pie/sector chart)

    1. Number of pets owned by each student in a class of 28 students (values range from 0 to 6).
    2. Percentage of students who chose each of four different sports as their favourite.
    3. Heights (cm) of 30 students to show the overall shape of the distribution.
    4. Temperature recorded every hour over a 24-hour period.

  3. Full Analysis

    Number of books read by 15 students over the school holidays: {4, 7, 2, 9, 5, 3, 7, 1, 8, 6, 7, 4, 5, 3, 6}

    1. Write out the stems and leaves for a stem-and-leaf plot of this data.
    2. Calculate the mean, median, mode, and range.
    3. Write three conclusions about reading habits during the holidays based on your analysis.

  4. Sort and Calculate

    Sort each dataset and find mean, median, mode, and range.

    1. Dataset D — Daily steps over 8 days: {6 200, 8 400, 5 900, 7 100, 8 400, 9 000, 6 800, 8 400}
    2. Dataset E — Marks on 10 spelling tests: {14, 16, 20, 18, 14, 17, 20, 14, 19, 16}
    3. Dataset F — Ages of members of a chess club: {12, 14, 13, 15, 12, 14, 16, 12, 13, 14, 15, 12}

  5. Which Measure of Centre?

    For each dataset, state which measure of centre best summarises it. Explain your choice.

    1. House prices: {$420 000, $450 000, $380 000, $410 000, $1 800 000, $430 000, $400 000}
    2. Shoe sizes in a class of 25 students.
    3. 1 km run times: {5.2, 5.8, 6.1, 5.5, 5.9, 6.3, 5.7, 5.4, 5.6, 6.0}
    4. Number of siblings each student has: {0, 1, 2, 1, 0, 3, 1, 0, 2, 1, 0, 1, 4, 1, 0}

  6. Outliers and Their Effect

    1. Dataset: {4, 5, 6, 5, 7, 6, 5, 4, 6, 50}. Calculate the mean, median, and mode with and without the outlier (50). Which measure is most affected?
    2. A student scores {72, 68, 75, 74, 71, 70, 69, 3} on 8 tests (3 was because she was sick). Calculate the mean with and without the score of 3. How does the mean change?
    3. Explain in your own words why the median is often preferred when a dataset contains an outlier.
    4. Give a real-world example where an outlier would make the mean a misleading summary statistic.

  7. Stem-and-Leaf Plots

    1. Draw a stem-and-leaf plot for: {34, 41, 27, 38, 52, 29, 45, 33, 48, 56, 31, 44, 37, 50, 26}
    2. Use your plot to find the median and mode.
    3. Calculate the mean and range for this dataset.
    4. Using the plot, identify the stem (tens digit) in which most values appear.
    5. Read the following plot and list all the original data values:
      2 | 1 4 8
      3 | 0 3 5 9
      4 | 2 6
      5 | 1 7
      Key: 2|1 = 21
    6. For the plot in (e), find the median, mode, and range.

  8. Compare Two Classes

    Test scores (out of 50) for two classes:

    Class A (leaves) Stem Class B (leaves)
    9 8 5 2 2 3 6
    7 6 4 1 3 1 4 5 8
    8 5 3 0 4 2 4 7
    2 5 0

    Key: for Class A read leaf then stem (e.g. leaf 2, stem 5 = 52)  •  for Class B read stem then leaf (e.g. stem 5, leaf 0 = 50)

    1. List all scores for Class A and Class B.
    2. Calculate the mean and range for each class.
    3. Find the median for each class.
    4. Which class performed better? Use statistics to justify your answer.
    5. Which class had more spread in their results? Explain.

  9. Interpreting a Column Graph

    How Year 7 students travel to school:

    Method WalkBusCarTrainCycle
    Number of students 2845371812
    1. How many students are represented in total?
    2. What percentage of students travel by bus? (Round to 1 decimal place.)
    3. What fraction of students travel by car or walk? Express as a simplified fraction.
    4. Which two methods combined account for more than half of all students?
    5. A student says the mode of this data is "Bus." Is she correct? Explain what mode means in this context.
    6. Write two conclusions about how Year 7 students travel to school, using the data to support each.

  10. Full Statistical Investigation

    Minutes per day spent reading for pleasure (20 Year 7 students):

    {0, 15, 30, 20, 45, 10, 0, 25, 60, 15, 30, 20, 10, 45, 0, 30, 20, 15, 40, 25}

    1. Sort the data and construct a stem-and-leaf plot (use stems 0, 1, 2, 3, 4, 5, 6).
    2. Calculate the mean, median, mode, and range. Show all working.
    3. The researcher says: "Most Year 7 students don't read at all for pleasure." Does the data support this claim? Explain using specific statistics.
    4. Identify any potential outliers and explain how they affect the mean.
    5. Write a brief report (4–5 sentences) summarising the reading habits of this group. Use measures of centre and spread.
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