The data show a positive association:

students who studied longer tended to have higher scores in this illustrative data set.

But this does not automatically establish causation.

---

Correlation and Causation

Suppose your data show:

more sleep ↔ higher test scores

You should not automatically conclude:

more sleep caused higher test scores.

Other variables may include:

• study habits
• stress
• attendance
• health
• workload
• motivation

If your investigation is observational, a more appropriate conclusion is:

"Greater reported sleep duration was associated with higher test scores in the sample."

This describes what the evidence actually shows.

---

Using Probability in a Capstone Project

A capstone investigation can also focus on probability.

For example:

"Does this spinner behave according to its theoretical probabilities?"

Suppose a spinner has four equal sections.

The theoretical probability of each colour is:

1/4 = 25%

If it is spun 200 times, the expected frequency for each colour is:

200 × 1/4 = 50

Students can compare theoretical and experimental results.

---

Theoretical and Experimental Probability

Suppose 200 spins produce:

The results are not exactly equal.

That is expected because of random variation.

Experimental probability for red:

54/200 = 0.27 = 27%

Theoretical probability:

25%

The difference alone does not prove that the spinner is biased.

---

Expected Value Investigations

Another possible project is to design or evaluate a game.

Suppose a game costs:

$2 to play

Possible outcomes are:

Expected winnings:

E(X) = 10(1/10) + 3(2/10) + 0(7/10)

= 1 + 0.60

= $1.60

Expected net gain:

1.60 − 2.00

= −$0.40

Therefore, the player expects to lose:

$0.40 per play in the long run.

This could form the basis of an investigation into whether a game is mathematically fair.

---

Comparing Theoretical Predictions with Data

Probability projects can combine theory and experimentation.

For example:

1. Calculate theoretical probabilities.
2. Predict expected frequencies.
3. Conduct the experiment.
4. Record observed frequencies.
5. Calculate experimental probabilities.
6. Compare theoretical and experimental results.
7. Explain differences using random variation.
8. Evaluate whether the number of trials was sufficient.

This connects probability directly with statistical investigation.

---

Looking for Outliers

An outlier should be investigated rather than automatically removed.

Suppose most travel times are:

15–50 minutes

but one student reports:

180 minutes

Possible explanations include:

• the student genuinely travels a very long distance
• the student misunderstood the question
• the value was entered incorrectly
• unusual circumstances affected the journey

The correct response is not simply:

"Delete the outlier."

Instead ask:

Why is it unusual, and how does it affect the analysis?

---

Comparing Results With and Without an Outlier

Consider:

10, 12, 13, 14, 15, 16, 18, 80

Mean:

178/8 = 22.25

Median:

(14 + 15)/2 = 14.5

The value 80 has strongly increased the mean.

If 80 is a genuine observation, this is important information about the distribution.

The median may provide a better measure of a typical value, but the unusual observation should still be discussed.

---

Drawing Evidence-Based Conclusions

A conclusion should answer the original research question using actual evidence.

Weak conclusion:

"Students don't sleep much."

Better conclusion:

"The median reported sleep duration among the 120 sampled students was 6.9 hours, while the mean was 6.8 hours."

Stronger:

"The median reported sleep duration among the 120 sampled students was 6.9 hours. About 68% reported fewer than eight hours, suggesting that sleep durations below eight hours were common in this sample."

The conclusion becomes stronger because it refers directly to evidence.

---

Separate Results from Interpretation

It is useful to distinguish between:

Result:
"The median travel time was 31 minutes."

and:

Interpretation:
"This suggests that a typical student in the sample spent about half an hour travelling to school."

and:

Conclusion:
"The sample therefore provides evidence that travel times of approximately 30 minutes are typical among the students represented by the investigation."

These statements are related, but they perform different roles.

---

Avoid Overgeneralization

Suppose you survey 30 students from one Grade 10 class.

Your evidence applies directly to:

those 30 students.

It may provide some information about other Grade 10 students, but the ability to generalize depends on how representative the sample is.

It would be inappropriate to claim:

"Teenagers everywhere sleep 6.8 hours."

The conclusion must remain within the limits of the evidence.

---

Communicating Findings with Tables

Tables are useful when readers need exact values.

For example:

A table gives precise numerical information.

A graph can then help readers see the overall pattern.

---

Communicating Findings with Graphs

A good graph should be:

accurate

clearly labelled

easy to interpret

appropriate for the data

directly relevant to the research question

Avoid:

• unnecessary 3-D effects
• misleading scales
• missing units
• excessive decoration
• too many unrelated graphs

The goal is communication, not decoration.

---

Combining Graphs and Statistical Summaries

Strong statistical communication often combines several forms of evidence.

For example:

Histogram

Shows the shape of the distribution.

Median

Describes a typical value.

IQR

Describes the spread of the middle half.

Sample size

Shows how much data were collected.

Written interpretation

Explains what these results mean.

No single graph or statistic usually tells the complete story.

---

Writing the Final Report

A complete capstone report can be organized into the following sections.

Research Question

Clearly state what you investigated.

Background and Purpose

Explain why the question is interesting or useful.

Method

Explain:

• population
• sample
• sampling method
• variables
• collection procedure

Data

Present organized data or an appropriate summary.

Analysis

Include appropriate:

• graphs
• tables
• measures of centre
• measures of spread
• percentages
• probability calculations
• comparisons
• relationships

Findings

Describe the important patterns.

Conclusion

Answer the research question using evidence.

Evaluation

Discuss strengths, weaknesses, uncertainty, and improvements.

---

Evaluating Strengths

A strong investigation might have:

• a clearly defined research question
• an appropriate sample
• sufficient data
• consistent measurements
• neutral survey questions
• appropriate statistical calculations
• suitable graphs
• conclusions supported by evidence
• transparent reporting

Do not simply write:

"My project was good."

Identify specific strengths.

For example:

"A strength of the investigation was the use of stratified sampling across six grade levels, which reduced the risk that one grade would dominate the sample."

---

Evaluating Limitations

Every real investigation has limitations.

Possible limitations include:

• small sample size
• convenience sampling
• voluntary response
• non-response
• inaccurate measurements
• self-reported data
• unclear survey questions
• limited time
• missing data
• uncontrolled variables
• measurement precision
• unusual observations
• restricted population

A limitation should explain how it could affect the conclusion.

Weak:

"The sample was small."

Better:

"Only 25 students participated, so random differences between individuals may have had a relatively large effect on the sample statistics."

---

Self-Reported Data

Many student investigations use self-reported information.

For example:

"How many hours did you sleep last night?"

A participant may:

• forget
• estimate incorrectly
• round the answer
• misunderstand the question
• report what seems socially acceptable

Therefore, self-reported data should often be identified as a limitation.

---

Bias

Bias is a systematic influence that can move results away from an accurate representation of the population.

Possible forms include:

• selection bias
• voluntary response bias
• non-response bias
• wording bias
• measurement bias

Identifying bias is not simply a criticism of the project.

It demonstrates understanding of the quality of the evidence.

---

Random Variation Versus Bias

These ideas should not be confused.

Random variation causes unpredictable differences from sample to sample.

Bias systematically pushes results in a particular direction.

Increasing sample size generally helps reduce sampling variability.

However, simply collecting more observations does not necessarily remove bias.

A sample of 10,000 badly selected people can still be biased.

---

Reliability and Validity

Reliability asks:

Would the method give reasonably consistent results if repeated?

Validity asks:

Does the method actually measure what we intended to investigate?

For example, suppose we want to measure physical fitness using only student height.

Height can be measured very reliably.

But height is not a valid measure of overall fitness.

A strong project considers both reliability and validity.

---

Uncertainty

Statistical conclusions are rarely perfectly certain.

Sources of uncertainty include:

• sampling variation
• measurement error
• random processes
• missing data
• model assumptions

Appropriate statistical language includes:

"The data suggest..."

"The sample provides evidence that..."

"There appears to be an association..."

"Within this sample..."

This is not weak writing. It accurately represents what statistical evidence can establish.

---

Improving the Investigation

After identifying a limitation, propose a realistic improvement.

Good evaluation connects each weakness to a practical improvement.

---

A Complete Example

Question

Is there an association between sleep duration and recreational screen time among Grade 10 students?

Plan

Population:

Grade 10 students at the school.

Sample:

80 students selected from several Grade 10 classes.

Variables:

• reported sleep duration in hours
• reported recreational screen time in hours

Collection

Students complete an anonymous questionnaire.

Organization

Responses are entered into a spreadsheet with one student per row.

Analysis

Calculate:

• mean and median sleep duration
• mean and median screen time
• range and IQR
• scatter plot
• possible correlation pattern

Possible Finding

Students reporting higher screen time tend to report somewhat lower sleep duration.

Appropriate Conclusion

"The sample showed a negative association between reported recreational screen time and reported sleep duration."

Inappropriate Conclusion

"Phones cause students to sleep less."

The investigation is observational and cannot establish that causal claim by itself.

Limitations

• self-reported data
• possible non-response
• other variables were not controlled
• data came from one school
• screen-time estimates may be inaccurate

Improvements

Future investigations could use:

• larger samples
• data from several schools
• device-recorded screen time
• sleep tracking
• additional variables such as workload or exercise

This turns a simple data collection exercise into a complete statistical investigation.

---

Capstone Project Workflow

A practical workflow is:

Choose a meaningful statistical question.

Plan the population, sample, variables, and method.

Predict what you think the data might show, if appropriate.

Collect the data carefully.

Check the data for errors and missing values.

Organize the data into useful tables.

Calculate appropriate statistical summaries.

Visualize the data using suitable graphs.

Analyze patterns, differences, distributions, or relationships.

Interpret what those patterns mean.

Conclude using evidence.

Evaluate strengths, limitations, uncertainty, and bias.

Improve the investigation by suggesting realistic changes.

Present the findings clearly.

---

Suggested Capstone Project Checklist

Before submitting a project, check that you can answer yes to most of these questions:

• Is my research question clearly stated?
• Is it genuinely statistical?
• Have I identified my population?
• Have I explained how my sample was selected?
• Have I identified my variables?
• Did I collect relevant data?
• Did I organize and check my data?
• Did I use appropriate statistical summaries?
• Did I choose appropriate graphs?
• Are my graphs correctly labelled?
• Did I interpret the graphs rather than simply include them?
• Did I answer the original research question?
• Did I support my conclusion with numerical evidence?
• Did I distinguish correlation from causation?
• Did I discuss possible bias?
• Did I identify limitations?
• Did I suggest realistic improvements?
• Are my conclusions no stronger than the evidence allows?

---

Common Mistakes

Starting without a clear research question

Collecting random data and deciding what to investigate afterward often produces a weak project.

Collecting data that do not answer the question

Every variable should have a purpose.

Using a biased sample

Convenient data are not necessarily representative data.

Using every possible statistical calculation

More calculations do not automatically create a better analysis. Choose statistics that help answer the question.

Choosing inappropriate graphs

Match the graph to the type of data and the purpose of the analysis.

Reporting graphs without interpreting them

Explain what the patterns actually mean.

Ignoring spread

Two groups can have similar averages but very different variability.

Deleting outliers automatically

Investigate unusual observations before deciding how to treat them.

Confusing association with causation

Observational data generally cannot establish causation by themselves.

Making claims beyond the population studied

A small sample from one school does not necessarily represent all teenagers.

Hiding inconvenient results

A statistical investigation should report the evidence honestly, even when the results do not support the original prediction.

Ignoring limitations

Recognizing limitations strengthens the investigation because it demonstrates critical statistical reasoning.

---

Key Terms

Capstone project — a culminating investigation that combines multiple skills and concepts into a complete study.

Statistical investigation — a systematic process of answering a question using data.

Research question — the specific question an investigation attempts to answer.

Population — the complete group about which conclusions are intended.

Sample — the subset of the population from which data are collected.

Representative sample — a sample that reasonably reflects important characteristics of the population.

Variable — a characteristic that can take different values.

Categorical data — observations classified into groups or categories.

Numerical data — observations represented by numerical measurements or counts.

Frequency — the number of times an outcome or category occurs.

Relative frequency — the proportion of observations belonging to an outcome or category.

Mean — the sum of numerical observations divided by their number.

Median — the middle observation in ordered data.

Mode — the most frequently occurring value or category.

Range — the difference between the maximum and minimum.

Interquartile range (IQR) — Q₃ − Q₁, describing the spread of the middle 50% of observations.

Distribution — the overall pattern of values in a data set.

Outlier — an observation unusually distant from the rest of the data.

Correlation — a statistical association between variables.

Expected frequency — the number of occurrences predicted from a probability model over a specified number of trials.

Expected value — the probability-weighted average numerical outcome of a random process.

Bias — a systematic influence that can distort results.

Random variation — natural unpredictable differences produced by sampling or random processes.

Reliability — the consistency of measurements or procedures.

Validity — the extent to which an investigation measures what it intends to measure.

Limitation — a feature of an investigation that restricts the strength or generalizability of its conclusions.

---

Key Takeaways

• A capstone data project combines the complete statistical investigation process in one independent study.
• Strong investigations begin with specific, measurable statistical questions.
• Planning should occur before data collection.
• The population, sample, sampling method, and variables should be clearly identified.
• Representative sampling is important when making conclusions about a population.
• Data should be collected consistently and checked before analysis.
• Statistical tools should be selected according to the question and type of data.
• Measures of centre describe typical values, while measures of spread describe variability.
• Graphs and tables should clarify the evidence rather than simply decorate the presentation.
• Scatter plots can reveal associations, but association alone does not establish causation.
• Probability investigations can compare theoretical probabilities, expected outcomes, and experimental results.
• Conclusions should answer the original question and refer directly to numerical or graphical evidence.
• Strong conclusions acknowledge uncertainty.
• Limitations should explain how weaknesses could affect the results.
• Improvements should directly address identified limitations.
• A successful capstone project demonstrates not only the ability to calculate statistics but the ability to use data to reason, evaluate evidence, and communicate a justified conclusion.

---

Check Your Understanding

1. Explain why a capstone data project should begin with a research question rather than simply collecting interesting data.
2. Rewrite "Are teenagers unhealthy?" as a specific statistical question that could realistically be investigated.
3. A student wants to investigate travel time among all students at a school but surveys only 20 students from their homeroom. Identify a limitation and suggest an improvement.
4. Explain why increasing the size of a strongly biased sample does not necessarily solve the problem.
5. For each variable, classify it as categorical, discrete numerical, or continuous numerical: transportation method, number of siblings, height, favourite subject, daily screen time.
6. A data set contains 12, 14, 15, 16, 16, 17, 18, 82. Explain why the mean and median may give very different impressions of a typical value.
7. Explain when you would choose a bar graph, histogram, box plot, and scatter plot.
8. Two groups have the same median but very different IQRs. What does this tell you about the groups?
9. A scatter plot shows that students who report more exercise also report higher happiness scores. Write an appropriate statistical conclusion that does not incorrectly claim causation.
10. A spinner is designed so that P(red) = 0.40. Calculate the expected number of red results in 500 spins. Explain why the actual number does not have to equal this value exactly.
11. A student finds an unexpected outlier and immediately deletes it. Explain why this is poor statistical practice.
12. Explain the difference between random variation and bias. Which problem can usually be reduced by increasing the sample size?
13. Design a capstone investigation involving two numerical variables. State your research question, population, sample, variables, collection method, appropriate graph, and at least two statistical summaries you would use.
14. Suppose your results do not support your original prediction. Explain what you should do when writing the final report.
15. Imagine that you have completed your capstone project. Write an outline for the final presentation showing how you would communicate your question, method, data, graphs, statistical analysis, conclusion, strengths, limitations, and possible improvements.
16.