Crash Course Statistics: Comparing Non-Standardized Scores

Key Concepts

• Statistics: Useful for comparing datasets (e.g., GPA, time spent on social media).
• Standardization: Important for comparing different measurements (e.g., SAT vs. ACT).

Comparing Different Tests

• Example: SAT (out of 1600) vs. ACT (out of 36).
• Both tests measure college readiness but use different scales, making direct comparison difficult.
• Standardization Process:
  1. Center the distribution around zero (subtract the mean from each score).
  2. Adjust further by measuring distance from the mean in terms of standard deviation.

Z-Scores

• Definition: A z-score indicates how many standard deviations a score is from the mean.
  • Z-score of 1 = 1 standard deviation above mean.
  • Z-score of -1 = 1 standard deviation below mean.
• Example:
  • Tony: SAT score of 1200, Mean = 1000, SD = 200 → Z-score = (1200-1000)/200 = 1.
  • Maia: ACT score of 25, Mean = 21, SD = 4.8 → Z-score = (25-21)/4.8 = 0.83.
• Interpretation: Z-scores allow for effective comparison of different test scores.

Percentiles

• Definition: Percentiles indicate the percentage of scores below a certain score.
  • Example: 83rd percentile means 83% of scores are lower than yours.
• Z-Distribution: Allows for easy calculation of percentiles based on z-scores.
• Example: To qualify for a gaming convention, you need to be in the 95th percentile.
  • Mean score = 2000, SD = 300, Z-score for 95th percentile ≈ 1.65.
  • Conversion back to raw score: 2,495.

Probability and Rarity

• A z-score can also indicate probability (e.g., being at the 90th percentile for marathon runners).
• Extreme Scores: Determining how "extreme" a score is can be subjective; varies by context.

Thought Bubble Example

• Game Booth Scenario: Comparing a weight of 270g against a mean of 200g with SD of 20g.
  • Z-score calculation: (270-200)/20 = 3.5.
  • 270g is larger than 99.99% of apples, indicating it’s likely something other than an apple.

Comparisons Beyond Tests

• Z-scores can help compare unrelated data, such as athlete performance across different sports.
• Example: Comparing Lebron James' points per game vs. Tom Brady's touchdowns per game to identify the GOAT.
• Importance of z-scores: They provide a relative measure of performance despite different scales.