Understanding NBA Salaries Through Statistics

Sep 16, 2024

Lecture Notes: Measures of Center in NBA Salaries

Overview

  • Today's activity focuses on understanding measures of center using NBA salaries as an example.
  • Previously covered measures of spread such as standard deviation.

Key Concepts

Measures of Center

  • Distribution: A representation or picture of a variable (e.g., NBA salaries).
  • Center of Distribution: The middle of the data, which can be represented by different measures of center.

Decision-Making Scenario

  • Consideration of a college basketball player dropping out of college based on financial prospects in the NBA.
    • Opinions may vary: stay in college for long-term benefit vs. grab the opportunity for immediate financial gain.
    • Statistics help make informed decisions based on data.

Measures of Center: Mean vs Median

Definitions

  • Median: The middle value in a list of numbers.
    • Stack numbers from smallest to largest, the median is the middle point.
  • Mean: The average of numbers.
    • Sum all numbers and divide by the count.

Characteristics

  • Median: Not affected by outliers; gives a true central tendency for skewed data.
  • Mean: Easier to calculate but is affected by outliers and skewed data.
    • Can be misleading if data has outliers or is not symmetric.

Analysis of NBA Salaries

  • Dot Plot Analysis: Visual representation of NBA salaries.
    • Example: James Harden earns $28 million vs. Chris Johnson $25,000.
  • Distribution Shape: Identified as skewed right due to a long tail to the right.
    • Outliers like James Harden’s salary can skew the mean.

Estimating Salaries

  • Typical Salary Range: Between $0 to $2.5 million.
  • Median Salary: $1.6 million.
  • Mean Salary: Over $5 million.

Comparing Mean and Median

  • Percentage Calculation: Estimating percentage of players earning above the mean.
    • Only about 28% earn more than the mean.
  • Above the Median: Median divides the dataset into two equal halves, about 50% above and below.

Misleading Claims

  • Use of mean to represent "typical" salaries can be misleading due to skewness and outliers.
  • Median is a more appropriate measure for skewed data as it is resistant to outliers.

Application Examples

  • Income in Santa Barbara: Right-skewed distribution due to a few high-income earners.
  • GPAs at UCSB: Left-skewed distribution due to few low scores.
  • Body Temperatures: Normally distributed, symmetrical.

Summary of Relationship

  • If data is skewed right, mean > median.
  • If data is skewed left, mean < median.
  • If data is symmetrical, mean ≈ median.

Important Takeaways

  • Median: Resistant to outliers and skewed data.
  • Mean: Not resistant to outliers; should be used with caution in skewed distributions.

Final Thoughts

  • Be cautious of misleading statistical claims.
  • Use median for better accuracy in skewed distributions.

Note: Practice further to internalize these concepts.