Capstone Projects

You've learned the theory. Now it's time to apply it.

This capstone lesson presents real-world statistical challenges where you'll integrate everything you've learned:

• Descriptive statistics and visualization
• Probability and distributions
• Hypothesis testing and confidence intervals
• Correlation, regression, and causation
• Critical thinking and fallacy detection

Choose one or more projects below. Work through them systematically, documenting your reasoning.

Goal: Explore a dataset, summarize findings, and make evidence-based conclusions.

Data sources:

• Kaggle datasets (kaggle.com/datasets)
• UCI Machine Learning Repository
• data.gov (government data)
• FiveThirtyEight data (github.com/fivethirtyeight/data)
• Your own data (work, hobby, research)

Required steps:

1. Data Exploration

• How many observations? Variables?
• What types of data? (categorical, numerical, time series)
• Missing values? Outliers?
• Create histograms, box plots, scatter plots

2. Descriptive Statistics

• Central tendency: mean, median, mode
• Spread: standard deviation, IQR
• Check distribution shape: normal? skewed?

3. Research Question

• Formulate a specific, testable question
• Example: "Do SAT scores differ by geographic region?"

4. Statistical Analysis

• Choose appropriate test (t-test, correlation, regression, chi-square)
• Check assumptions (normality, independence, sample size)
• Calculate test statistic and p-value
• Construct confidence intervals

5. Interpretation

• What do the results mean in plain English?
• Statistical vs practical significance?
• Limitations and confounders?
• Alternative explanations?

6. Visualization

• Create publication-quality graphs
• Tell a story with data
• Avoid misleading visualizations

7. Write-up

• Introduction (question and why it matters)
• Methods (data source, sample size, tests used)
• Results (numbers, tables, figures)
• Discussion (interpretation, limitations, future directions)

Goal: Design an experiment, collect data (or simulate), and analyze results.

Scenario: You run a website and want to increase signups.

Steps:

1. Hypothesis "Changing the signup button from blue to green will increase signup rate."

2. Design

• Control: Blue button
• Treatment: Green button
• Metric: Signup rate (%)
• Randomization: 50/50 split of visitors

3. Sample Size Calculation

• Current signup rate: 5%
• Minimum detectable effect: +1 percentage point (to 6%)
• Significance level: α = 0.05
• Power: 0.80
• Calculate required sample size using formulas or online calculator

4. Data Collection (can simulate)

• Generate simulated data with realistic effects
• Include some noise and variability

5. Analysis

• Calculate signup rates for both groups
• Two-proportion Z-test
• Confidence interval for difference
• Effect size and practical significance

6. Conclusion

• Is the result statistically significant?
• Is the improvement worth implementing?
• Cost-benefit analysis

7. Report

• Present to stakeholders (pretend)
• Recommendation: Roll out green button? More testing needed?

Extensions:

• Multivariate test (test multiple changes simultaneously)
• Sequential testing (analyze as data arrives)
• Account for multiple testing if running many tests

Goal: Find a statistical claim in the wild and critically evaluate it.

Sources:

• News articles
• Social media
• Advertisements
• Political claims
• Health/wellness products

Required analysis:

1. Identify the Claim "Drinking green tea burns 500 extra calories per day!"

2. Find the Source

• Original study? Or just marketing?
• Sample size? Study design?
• Peer-reviewed? Replicated?

3. Critical Questions

• Is this correlation or causation?
• What's the baseline / control group?
• Relative vs absolute effect?
• Cherry picking? Publication bias?
• Funding source / conflicts of interest?
• Sample representative?

4. Alternative Explanations

• Confounding variables?
• Reverse causation?
• Measurement error?
• Regression to the mean?

5. Calculate True Effect

• If they report relative risk, find absolute risk
• If they say "statistically significant," find effect size
• Compare claimed effect to plausible reality

6. Write-up

• Original claim (with source)
• Your analysis (with evidence)
• Conclusion: True? Exaggerated? False?
• Corrected interpretation

Examples to consider:

• Diet / supplement claims
• Political polls (methodology and interpretation)
• Vaccine/health scares
• Financial advice ("this strategy beats the market")
• Product effectiveness claims

Goal: Analyze a business problem using statistics and present actionable insights.

Sample scenarios:

Scenario A: Customer Retention

• Problem: Customer churn increasing
• Data: Customer demographics, usage patterns, churn status
• Analysis: What factors predict churn? (logistic regression)
• Recommendation: Target interventions for high-risk customers

Scenario B: Pricing Optimization

• Problem: What price maximizes revenue?
• Data: Historical sales at different prices
• Analysis: Price elasticity, demand curves, confidence intervals
• Recommendation: Optimal price range

Scenario C: Quality Control

• Problem: Defect rate seems high
• Data: Defect counts over time
• Analysis: Control charts, hypothesis tests vs target rate
• Recommendation: Process changes needed?

Required components:

1. Executive Summary

• Problem statement
• Key findings (2-3 bullet points)
• Recommendation (actionable)

2. Data & Methods

• Data sources and sample size
• Variables analyzed
• Statistical methods used
• Assumptions and limitations

3. Results

• Tables and visualizations
• Statistical tests with interpretation
• Confidence intervals
• Sensitivity analysis

4. Discussion

• Practical significance
• Limitations and caveats
• Risks and uncertainty
• Implementation considerations

5. Recommendations

• Clear, actionable next steps
• Expected impact (with uncertainty ranges)
• Monitoring plan

Goal: Write for non-statistical stakeholders. Avoid jargon. Focus on business impact.

Goal: Use simulation to understand a statistical concept or solve a problem.

Option A: Central Limit Theorem

• Start with a non-normal distribution (exponential, uniform, etc.)
• Sample from it repeatedly, calculate means
• Plot distribution of means
• Show how it becomes normal as n increases
• Demonstrate E[X̄] = μ and SE = σ/√n

Option B: Bootstrap Confidence Intervals

• Take a sample of data
• Bootstrap resample 10,000 times
• Calculate statistic (mean, median, correlation) for each
• Construct 95% CI using percentile method
• Compare to theoretical CI

Option C: Power Analysis

• Simulate studies with varying sample sizes
• Generate data under H₁ (effect exists)
• Test H₀ (no effect)
• Calculate proportion of times p < 0.05 (power)
• Show how power increases with n and effect size

Option D: P-Hacking Demonstration

• Simulate 20 studies testing a null effect
• Show that ~1 will have p < 0.05 by chance
• Demonstrate dangers of selective reporting

Option E: Type I vs Type II Errors

• Simulate data under H₀ and H₁
• Vary significance level (α) and sample size
• Calculate Type I error rate (false positives)
• Calculate Type II error rate (false negatives)
• Show the tradeoff

Tools: Python (NumPy, SciPy, Matplotlib), R, or even Excel with random number generation.

Deliverable: Annotated code + visualizations + explanation of what you learned.

You've built a solid foundation in statistics. Where to go from here?

Deepen your knowledge:

• Bayesian statistics: Probability as degree of belief
• Machine learning: Prediction and pattern recognition
• Causal inference: Going beyond correlation
• Time series forecasting: ARIMA, exponential smoothing
• Survey design and sampling: How to collect data properly
• Experimental design: Factorial designs, blocking, interactions

Practice continuously:

• Analyze datasets regularly (Kaggle, personal projects)
• Read research papers critically
• Follow data science blogs and journals
• Participate in competitions (Kaggle, DrivenData)

Learn tools:

• Python: pandas, NumPy, SciPy, statsmodels, scikit-learn
• R: tidyverse, ggplot2, statistical tests
• SQL: Data extraction and manipulation
• Visualization: Tableau, matplotlib, seaborn, ggplot2

Teach others:

• Explain concepts to reinforce understanding
• Write blog posts or tutorials
• Help peers with statistics questions

Stay skeptical:

• Question statistical claims
• Look for fallacies in the wild
• Demand evidence and good methodology
• Update beliefs based on evidence

The journey continues. Statistics is a skill that deepens with practice and never stops being useful.