Introduction to Statistics

Lecture Overview

• Lecturer: Justin Zeltzer from zstatistics.com
• Objective: Explain statistics in under half an hour without using math
• Target Audience: Beginners enrolled in a statistics course or those curious about statistics
• Theme for Examples: NBA basketball

Types of Data

Categorical Data

• Definition: Data that can be divided into categories
• Subtypes:
  • Nominal: No intrinsic order (e.g., 'What team does Steph Curry play for?')
  • Ordinal: Some kind of order (e.g., 'What position does Steph Curry play?')

Numerical Data

• Definition: Data that represents numbers
• Subtypes:
  • Discrete: Finite values (e.g., 'How many free throws has Steph missed?')
  • Continuous: Infinite subdivisions (e.g., 'What is Steph's height?')

Special Case: Proportions

• Example: 'What is Steph's three-point percentage this season?'
• Explanation: Proportions are essentially aggregates of nominal data providing a numerical summary.

Distributions

Types of Distributions

• Normal Distribution (Bell Curve): Common in statistics; bulk of data is in the middle.
• Other Distributions:
  • Uniform Distribution: Equal probability across values
  • Bi-modal Distribution: Two peaks
  • Skewed Distribution: Long tail on one side (e.g., left-skewed, right-skewed)

Probability Density Function (PDF)

• Describes the distribution of all players’ heights in the NBA
• Symmetrical with most players in the middle

Sampling Distributions

• Concept: If we sample, what is the probability distribution of their average height?
• Key Point: Larger sample sizes reduce variance, making extreme sample means less likely.

Sampling and Estimation

Sample Statistics vs. Population Parameters

• Example: Steph Curry's three-point percentage (sample statistic)
• Theta (θ): Represents the true, long-term value we are trying to estimate
• Confidence Intervals: Quantify uncertainty in our estimates

Example: Confidence Intervals

• Steph Curry: More data, narrower confidence interval
• Meyers Leonard: Less data, wider confidence interval

Parameters and Sample Statistics

Common Parameters

• Mu (μ): Mean of a numerical variable
• Sigma (σ): Standard deviation of a numerical variable
• Pi (π): Proportion of a categorical variable
• Rho (ρ): Correlation between two variables
• Beta (β): Gradient between two variables
• Theta (θ): General parameter

Sample Statistics Notation

• x-bar (x̄): Sample mean
• s: Sample standard deviation
• p: Sample proportion
• r: Sample correlation
• b: Sample gradient

Hypothesis Testing

Concept

• Example: Is Meyers Leonard shooting above 50%?
• Null Hypothesis (H0): An assumption to test against e.g., θ ≤ 0.5
• Alternative Hypothesis (H1): What we are seeking evidence for e.g., θ > 0.5
• Rejection Region: If sample statistic falls in this, null hypothesis is rejected.

Key Points

• P-value: Measures how extreme the sample is
• Level of Significance (usually 5%): If p-value is less than this, reject the null hypothesis
• Interpretation: Never use the words 'prove' or 'accept' in hypothesis testing conclusions

P-Hacking

Concept

• Misuse of p-values by testing multiple hypotheses and reporting only favorable outcomes
• Good Research: Theorize an effect and test it specifically
• Bad Research: Collect data first and then test for various effects (leading to false positives)

Real-World Implications

• P-hacking leads to invalid scientific research
• Proper methodology is crucial for valid conclusions

Conclusion

• Wrap-Up: Summarized the key points of statistics without intensive math
• Additional Resources: More in-depth discussions and videos available on zstatistics.com