Introduction to Statistics Lecture Notes

Introduction

• Presenter: Justin Zeltser from zstatistics.com
• Challenge: Explain statistics in under half an hour without maths.
• Goal: Develop intuition about statistics for beginners or the curious.
• Examples themed around the NBA.

Types of Data

• Categorical Data
  • Nominal: No order (e.g., NBA teams)
  • Ordinal: Some order (e.g., player positions)
• Numerical Data
  • Discrete: Countable values (e.g., number of free throws missed)
  • Continuous: Any value within a range (e.g., player's height)
• Proportions: Aggregates nominal data into numerical summaries.

Distributions

• Probability Density Function: Describes distribution of a population.
• Normal Distribution: Commonly occurs; mean-centered with rarer extremes.
• Other Distributions
  • Uniform Distribution: Equal probability across values.
  • Bimodal Distribution: Two peaks in data.
  • Skewed Distribution: Longer tail on one side (left or right skew).

Sampling Distributions

• Sampling: Selecting a group to infer about a population.
• Distribution of Sample Means: Skinnier than population distribution.
• Variance Reduction: Larger samples yield less variance in sample means.

Estimation

• Sample Statistic: An estimate of a population parameter.
• Theta (θ): Symbol for an unknown parameter.
• Confidence Intervals: Indicate range where a parameter likely falls.
• Example: Comparing Steph Curry and Myers Leonard's three-point shooting.

Parameters and Estimation

• Common Parameters
  • Mu (μ): Mean
  • Sigma (σ): Standard deviation
  • Pi (π): Proportion
  • Rho (ρ): Correlation
  • Beta (β): Gradient (regression)
• Sample Statistics: Estimates for parameters (e.g., x-bar for mean).

Hypothesis Testing

• Hypothesis Test: Tests evidence against a null hypothesis (H0).
• Alternate Hypothesis: What you seek evidence for.
• Rejection Region: Area beyond which null hypothesis is rejected.
• Significance Level: Commonly set at 5%.

P-Values

• Definition: Measures how extreme a sample is.
• Interpretation: Lower p-value indicates more evidence against null hypothesis.
• Relation to Hypothesis Tests: If p-value < 0.05, likely in rejection region.

P-Hacking

• Definition: Manipulating data to find significant results.
• Problem in Research: Testing multiple hypotheses increases false positives.
• Good vs. Bad Research: Good research tests a single hypothesis; bad research tests many and reports only significant findings.

Conclusion

• Statistics deals with uncertainty and inference rather than proof.
• Importance of understanding potential for misuse in research (p-hacking).
• Additional resources and in-depth discussions available on zstatistics.com.