Lecture Notes: Measuring Data Position in a Data Set

Introduction to Data Positioning

• Focus on methods to measure the position of data relative to other values in a data set.

Percentiles

• Definition: Percentiles split a data set into 100 evenly divided pieces.
• Purpose: Indicates the relative position of a score within a data set.
• Example: Scoring in the 90th percentile means 90% of scores are below yours.
  • Does not indicate the actual score achieved.
  • Implies rank relative to other scores.

Understanding Percentile through an Example

• Scenario: Exam score in 90th percentile.
• Explanation: 90% of scores are lower than yours; 10% are higher.
• Notes: A high score (e.g., 95/100) or a relatively lower score (e.g., 72/100) can both be in the 90th percentile depending on other scores.

Calculating Percentiles

• Manual Calculation: Requires ordering and examining data values to find the specific percentile position.
• Using Software:
  • Use the quantile function in R.
  • Percentiles are a specific type of quantile.
  • Quantile: General term for data division into arbitrary pieces.
  • Function Usage in R:
    • X: List of data values.
    • probs: Relative position in decimal form (e.g., 90th percentile is 0.90).

Example in R: Calculating the 73rd Percentile

• Data Set: 29 ages of Academy award-winning actors.
• Procedure:
  1. Create a list called ages with the data values.
  2. Use quantile function with probs set to 0.73 for the 73rd percentile.
• Result: 73rd percentile is 63.32 years.
• Interpretation:
  • 63.32 years is greater than 73% of the ages.
  • Equivalent to saying 73% of ages are less than 63.32 years.
  • Also implies 63.32 is smaller than 27% of ages (completing the 100%).

Conclusion

• Percentiles provide meaningful insights into the rank of data within a data set.
• Useful for interpreting scores, ages, or other measurable data in different contexts.