Lecture Notes: Measures of Center - NBA Salaries
Overview
- Discussion focus: Salaries of NBA players as a vehicle to understand measures of center.
- Previous session: Measures of spread and standard deviation.
- Today's focus: Measures of center in distributions using NBA salaries as the variable of interest.
Key Concepts
Measures of Center
- Center of Distribution: The middle of a data set or "cloud of data."
- Mean: Average; calculated by summing all values and dividing by the number of values.
- Easy to calculate but sensitive to outliers and skewed data.
- Median: Middle point; half of the data points are below and half are above.
- Resistant to outliers and skewed data.
Discussion Exercise
- Scenario: A college basketball player considering dropping out for the NBA.
- Financial perspective decision.
- Opinions vary from staying in college for long-term security to taking immediate financial benefits.
- Data analysis is crucial to make informed decisions.
Data Analysis
Distribution of NBA Salaries
- Example of salary distribution for Texas NBA players.
- James Harden: Salary over $28 million, notable outlier.
- Chris Johnson: Salary $25,000, significant variability in player salaries.
- Distribution displays skewness, particularly right-skewed.
Understanding Skewness
- Skewed Right: Tail is on the right (more rare, high-value outliers).
- Outliers: James Harden is a significant outlier; other potential outliers include Chris Paul's salary.
Practical Analysis
- Typical Salary Estimation: Most common between $0 and $2.5 million.
- Mean vs. Median in Skewed Data:
- Mean is higher due to outliers pulling the average up.
- Median gives a better "typical" value in skewed data.
Exercises
Estimating Percentages
- Above the Mean: 28% of players have salaries above the mean.
- Above the Median: By definition, 50% of players have salaries above the median.
Misleading Statistics
- Recruiter using mean salary as typical; misleading because only 28% earn more than the mean.
- Mean is inflated by top earners.
Shape and Center Comparisons
- Income in Santa Barbara: Skewed right.
- GPAs at UCSB: Skewed left.
- Body temperatures: Symmetrical, bell-shaped, mean equals median.
Key Takeaways
- Skewed Right: Mean > Median.
- Skewed Left: Mean < Median.
- Symmetrical: Mean = Median.
- Median is a more reliable measure of center in skewed distributions because it is not affected by outliers.
Summary
- Median is robust to skew and outliers.
- Mean is not resistant to skew and outliers and should not be used in skewed data.
- Use statistical reasoning when interpreting data and be cautious of misleading averages presented in media or reports.
Note: Practice questions and real-world applications to solidify understanding were recommended at the end of the lecture.