Lecture: Introduction to Statistics by Justin Zeltzer

Overview

• Goal: Explain statistics in under half an hour without math
• Audience: Beginners in statistics; those interested in basic concepts
• Theme: Examples from the NBA basketball

Types of Data

Categorical Data

• Nominal: No order
  • Example: Teams Steph Curry can play for
• Ordinal: Ordered categories
  • Example: Positions in basketball (Guard, Forward, Center)

Numerical Data

• Discrete: Finite numbers
  • Example: Number of free throws missed
• Continuous: Infinite subdivisions
  • Example: Player’s height (e.g., 191.3 cm)

Proportions

• Proportions as Numerical Data:
  • Example: Steph Curry’s three-point percentage
  • Percentages are proportions out of 100
  • Built from nominal data (made/missed shots)
  • Discussion on whether proportions are discrete or continuous

Distributions

Probability Density Function (PDF)

• Example: Height distribution of NBA players (Isaiah Thomas 5’9”, Boban Marjanovic 7’3”)
• Normal Distribution (Bell Curve): Most players around average height
• Other Types:
  • Uniform Distribution: Equal probability across all values
  • Bimodal Distribution: Two peaks
  • Skewed Distribution: Long tail in one direction (left or right skew)

Sampling Distributions

• Concerned with the distribution of sample means rather than individual measurements
• Law of Large Numbers: Larger sample sizes yield average values closer to the population mean, reducing variance

Estimation and Inference

• Sample Statistic vs Parameter: Sample provides an estimate, not the precise population value
  • Example: Steph Curry’s three-point shooting percentage (Theta, Θ)
• Confidence Intervals: Provide a range where the true parameter likely falls
  • Individual player’s confidence intervals (e.g., Meyers Leonard vs Steph Curry)

Parameters and Statistics

• Parameters (Greek Letters):
  • Mu (μ): Mean
  • Sigma (σ): Standard deviation
  • Pi (π): Proportion
  • Rho (ρ): Correlation
  • Beta (β): Gradient in regression
• Sample Statistics (Roman letters):
  • X-bar (x̄): Sample mean
  • S: Sample standard deviation
  • P: Sample proportion
  • R: Sample correlation
  • B: Sample gradient

Hypothesis Testing

• Null Hypothesis (H₀): Assumes no effect or status quo
• Alternative Hypothesis (H₁): What we seek evidence for (e.g., player shooting above 50%)
• Example: Meyers Leonard’s three-point shooting
• Rejection Region: Area where sample is extreme enough to reject H₀
• Significance Level (α): Often set at 0.05 (5%)

Key Points on Hypothesis Testing

• Never say “prove” or “accept” in conclusions
• Conclusions vary on evidence to reject or not reject H₀
• Analogy with judicial system: not enough evidence ≠ innocence

p-Values

• Definition: Measures the extremity of the sample statistic
• Interpretation: Smaller p-value means more extreme and more evidence against H₀
• Relation to rejection region and significance level (α)

Ethical Considerations in Research

P-Hacking

• Definition: Manipulating data collection or analysis to achieve desired p-values
• Good Research: Theorize, collect, test targeted effect
• Bad Research: Collect data broadly, test many effects, report significant ones
• Problem: Increases likelihood of finding false positives

Additional Resources

• Further learning available at zstatistics.com

Conclusion

• Summary of key concepts with NBA examples
• Aim to provide intuition and understanding of statistical principles