Introduction to Normal Probability Distributions

Empirical Rule Limitations

• The empirical rule is useful for identifying likely and unlikely values.
• Limited to probability questions involving x-values that are exactly 1, 2, or 3 standard deviations from the mean.
• Cannot answer questions like: What is the probability that a man's foot length is more than 13 inches?

Normal Distribution Example: Male Foot Length

• Mean foot length = 11 inches.
• Standard deviation = 1.5 inches.
• We need to find the probability that x > 13 inches:
  • Calculate number of standard deviations 13 is from the mean.
  • 13 inches is 2 inches above the mean.
  • Number of standard deviations = 2 inches / 1.5 inches = 1.33.
  • This is known as finding the z-score.

Z-Score Calculation

• Formula: ( z = \frac{x - \mu}{\sigma} )
  • ( x ): value of interest
  • ( \mu ): mean
  • ( \sigma ): standard deviation
• Positive z-score: x is above the mean.
• Negative z-score: x is below the mean.

Example: 8.5-inch Foot Length

• Mean = 11 inches, SD = 1.5 inches.
• ( z = \frac{8.5 - 11}{1.5} = -1.67 )
• Negative z-score indicates x is below the mean.

Comparing Z-Scores Across Different Distributions

• Z-scores allow comparison of x-values from different distributions.
• Example: Male and Female Foot Length
  • Sam’s foot length = 13.25 inches.
  • Candice’s foot length = 11.6 inches.
  • Male mean = 11 inches, SD = 1.5 inches.
  • Female mean = 9.11 inches, SD = 1.2 inches.
  • Sam’s z-score = 1.5.
  • Candice’s z-score = 1.75.
  • Candice has a larger foot relative to other women than Sam does relative to other men.

Test Scores Example

• Maria’s class: Mean = 70, SD = 5.
• Lamar’s class: Mean = 85, SD = 10.
• Maria scored 76; Lamar scored 80.
• Maria’s z-score = 1.2 (above the mean).
• Lamar’s z-score = -0.5 (below the mean).
• Maria performed better relative to her class than Lamar did to his.

Key Takeaways

• Z-scores standardize data for comparison across different distributions.
• Positive z-scores indicate values above the mean, negative below.
• Useful for comparing performance or measurements relative to different groups.
• Remember to check whether the z-score is positive or negative to assess relative position.