Lecture on Measures of Relative Standing

Introduction

• Measures of Center & Variation: Previously discussed measures of center and variation.
  • Center: One number representing the center of a dataset.
  • Variation: One number representing the spread of the dataset.
• Measures of Relative Standing: How individual data points compare within a dataset.
  • Compare data points from different scales.

Example of Different Scales

• SAT Scores: Mean = 1518, Standard Deviation = 325
• ACT Scores: Mean = 21.1, Standard Deviation = 4.8
• Different scales make direct comparisons invalid.

Z-Score

• Definition: Indicates how many standard deviations a data point is from the mean.
• Formula:
  • For a sample: ( z = \frac{X - \text{mean}}{\text{standard deviation}} )
  • For a population: Use population symbols.
• Example Calculation:
  • Dataset: [10, ..., 5 total numbers]
  • Mean = 12.4, Standard Deviation = 3.05
  • Z-score for 10: ( z = \frac{10 - 12.4}{3.05} = -0.79 )
  • Interpretation: 10 is 0.79 standard deviations below the mean.

Comparing Different Scales with Z-Scores

• SAT vs. ACT Example:
  • SAT: Score = 1030, Z-score = -1.5
  • ACT: Score = 14, Z-score = -1.48
  • Conclusion: ACT score is better (higher z-score).

Usual vs. Unusual Values

• Usual Values: Z-scores between -2 and 2
• Unusual Values: Z-scores beyond ±2
• Examples:
  • 5 might have a z-score < -2
  • 1239 might have a z-score > 2

Percentiles

• Definition: Divides data into 100 groups.
• Example Calculations:
  • Value 41:
    • Count values < 41: 5 out of 28
    • Percentile: ( \frac{5}{28} \times 100 = 17.86 \approx 18 )
  • Value 89:
    • Count values < 89: 27 out of 28
    • Percentile: ( \frac{27}{28} \times 100 = 96.42 \approx 97 )
• Finding Data from Percentiles:
  • Example: 80th percentile of dataset
  • Formula: ( L = \frac{k}{100} \times n )

Quartiles

• Definition: Divides data into four groups.
• Quartiles:
  • Q1: 25%
  • Q2: 50% (Median)
  • Q3: 75%
• Example:
  • Q3 = same as P75
  • Calculate using location formula

Box and Whisker Plot

• Components:
  • Minimum, Q1, Median (Q2), Q3, Maximum
  • Visual representation on a number line
• Example:
  • Minimum = 20, Q1 = 45, Median = 60, Q3 = 71, Maximum = 89
  • Plot illustrates data spread and center

Conclusion

• Understanding Z-scores, percentiles, and quartiles help compare data points and analyze data distribution effectively.