Lecture on Z-Scores

Introduction to Z-Scores

• Definition: Z-scores are a way to express average and standard deviation.
  • A z-score of 1 means one standard deviation above the average.
  • A z-score of -1 means one standard deviation below the average.
• Normal Curve:
  • The average (mean, denoted as μ) is at the center of the normal curve with a z-score of 0.
  • Z-scores are placed on the normal curve:
    • +1 z-score: one standard deviation above the mean.
    • +2 z-score: two standard deviations above the mean.
    • -1 z-score: one standard deviation below the mean.
    • -2 z-score: two standard deviations below the mean.

Empirical Rule and Z-Scores

• 68% of data lies within one standard deviation (z-scores -1 to +1).
• 95% of data lies within two standard deviations (z-scores -2 to +2).

Importance of Z-Scores

• Allows calculation of area under the curve for any z-score (including non-integers like 1.5).
• Z-table: Used to find areas for specific z-scores.

Example Problem

• Given:
  • Test scores normally distributed
  • Mean (μ) = 75, Standard Deviation (σ) = 10
• Task: Find probability of a student scoring above 60.
• Steps to Solve:
  1. Place Mean on Curve: Mean of 75 is placed at the center.
  2. Identify Score of Interest: Score of 60 placed below mean.
  3. Calculate Z-Score for 60:
     • Formula: ( z = \frac{x - μ}{σ} )
     • Calculation: ( z = \frac{60 - 75}{10} = -1.5 )
  4. Determine Areas on Curve:
     • Section 1: Left of 60 (z = -1.5)
       • Use Z-table: Area = 0.0668 (or 6.68%)
     • Section 3: Right of mean (75)
       • Half of the curve: Area = 50%
     • Section 2: Between 60 and 75
       • Total area = 100%
       • Calculate: Area = 100% - 6.68% - 50% = 43.32%
  5. Calculate Probability Above 60:
     • Add area of Section 2 and Section 3
     • Probability = 43.32% + 50% = 93.32%

Conclusion

• Result: Probability a student scores above 60 is 93.32%.
• Next Topic: Overview of t-scores for cases where z-scores are not applicable.