Standardization and Z-Scores

Introduction

• Lecturer: Adriene Hill
• Topic: Comparing different measurements using standardization, specifically through the concept of z-scores.

Comparing Different Measurements

• Issue: Comparing measurements that aren't on the same scale (e.g., SAT vs. ACT scores).
• Example: College students’ test scores (SAT out of 1600, ACT out of 36 points).
• Objective: Make scores comparable even if they aren't measured the same way.

Process of Standardization

Step 1: Centering Around Zero

• Method: Subtract the mean of each test from each score.
• Purpose: Adjusts test scores so that a score of 0 is the mean for both tests.
• Example: SAT (1000) & ACT (21) adjusted to 0.

Step 2: Rescaling Using Standard Deviation

• Method: Divide adjusted scores by the standard deviation of each test.
• Purpose: Rescales to a standard deviation of 1 for both tests.
• Result: Produces a z-score.

Z-Scores

• Definition: Indicates how many standard deviations a point is from the mean.
  • Positive z-score: above the mean
  • Negative z-score: below the mean
• Example Calculation: Tony (SAT) & Maia (ACT)
  • Mean (SAT): 1000, Standard Deviation (SD): 200
    • Tony’s Score: 1200 -> z = (1200-1000)/200 = 1
  • Mean (ACT): 21, SD: 4.8
    • Maia’s Score: 25 -> z = (25-21)/4.8 = 0.83

Application: Percentiles and Comparisons

Percentiles

• Definition: Indicates the percentage of the population with lower scores.
• Example: Being in the 83rd percentile for height.
• Importance: Useful in understanding one's standing relative to a distribution.

Using Z-Scores to Find Percentiles

• Example: Gaming competition (Top 5% required)
  • Game Mean Score: 2000, SD: 300, Z-score for 95th percentile: 1.65
  • Required Score: (ZSD) + Mean -> 1.65300 + 2000 = 2495
  • Application: Transforms percentile requirements to concrete scores.

Interpreting Z-Score in Real-Life Scenarios

• Example: County fair game with apples vs. mystery items
  • Apple mean weight: 200g, SD: 20g
  • Mystery item weights 270g; Z-score = (270-200)/20 = 3.5
  • Interpretation: Object lies beyond usual range for apples (extremely large apple).

Conclusion: The Utility of Z-Scores for Comparability

• Key Point: Z-scores help make disparate data comparable.
• Applications: Assessing likelihoods (e.g., meeting celebrities, athletes’ performance).
• Example: Comparing athletes from different sports (e.g., Lebron James vs. Tom Brady).
• Final Thought: Standardization like z-scores can help identify the 'Greatest Of All Time' across different domains.