Lecture Notes: Understanding Measures of Center

Introduction

• Activity 1C: Discussing NBA players' salaries to understand measures of center.
• Previous Topic: Measures of spread, specifically standard deviation.
• Current Topic: Measures of center in distributions.

Key Concepts

• Distribution: A picture of a variable (e.g., NBA salaries).
• Center of Distribution: The middle of a data set.
  • Different measures have different advantages and disadvantages.

Discussion: Dropping Out of College

• Scenario: NBA prospect considering dropping out for financial reasons.
• Points of View:
  • Stay in college: An NBA career is temporary.
  • Drop out: Grab financial opportunity while possible.
• Statistics Approach: Let data guide the decision.

Measures of Center

• Median: Middle point of data.
  • Half the numbers are on either side when ordered.
  • Advantage: Not affected by outliers.
• Mean (Average): Add all numbers and divide by count.
  • Advantage: Easier to calculate.
  • Disadvantage: Affected by outliers and skewed data.

Analyzing NBA Salaries

• Distribution Shape: Skewed right.
• Outliers: Extremely high salaries like James Harden's.
• Typical Salary Estimate: Between $0 and $2.5 million.

Comparing Mean and Median

• Mean: $5.3 million.
• Median: $1.6 million.
• Difference: Mean is higher due to skew and outliers.

Percentage of Players with Salaries

• Above Mean: 28% of players.
• Above Median: 50% of players (by definition).
• Explanation: Skewness affects mean but not median.

Misleading Claims

• Recruiter's Statement: Claiming $5 million as typical is misleading.
  • Only 28% earn more than this.
  • Mean is not a good measure due to skew.
  • Median is more appropriate for skewed data.

Situations Analysis

• Income in Santa Barbara: Skewed right, mean is greater than median.
• GPAs at UCSB: Skewed left, mean is less than median.
• Body Temperatures: Symmetrical, mean equals median.

Summary

• Mean vs Median:
  • Skewed Right: Mean > Median
  • Skewed Left: Mean < Median
  • Symmetrical: Mean ≈ Median
• Robustness: Median is robust against outliers; mean is not.

Conclusion

• Use median for skewed distributions to avoid misleading interpretations.
• Understand how skew and outliers can manipulate data representation.
• Practice identifying misleading claims and the appropriate use of measures.