Module 13: Introduction to Normal Probability Distributions

Key Concepts

• Normal Probability Curves: The area within one standard deviation of the mean equals 68% (0.68).
• Probability of Values Outside One Standard Deviation: The leftover probability is calculated as 1 - 0.68 = 0.32.
  • This 0.32 is divided between the two tails of the distribution, meaning each tail has an area of 0.16.

Example: Probability of X > 1 SD

• Question: What is the probability that an X value is more than one standard deviation above the mean?
• Solution:
  • Total area under the curve = 1
  • Area within 1 SD = 0.68
  • Area outside 1 SD = 0.32 (0.16 in each tail)
  • Probability: 0.16 (for the tail to the right)

Example: Probability of Birth Weight < 100 Ounces

• Context: Birth weights follow a normal distribution with a mean of 120 ounces and a SD of 20 ounces.
• Calculation:
  • Mean = 120 ounces
  • 1 SD below mean = 100 ounces
  • Probability: Weight less than 100 ounces = Area left of 100 ounces = 0.16

Example: Probability of Birth Weight > 100 Ounces

• Context: Same normal distribution as previous example.
• Calculation:
  • Weights greater than 100 ounces include values greater than the mean and up to 1 SD above, plus additional tail.
  • Probability:
    • Area between 100 ounces and the mean = 0.68
    • Additional tail area (right of mean) = 0.16
    • Total Probability: 0.68 + 0.16 = 0.84 (84%)

Summary

• Key Takeaways:
  • Normal distribution probabilities can be calculated based on standard deviations.
  • Areas under the curve add up to 1.
  • Symmetry of the normal distribution allows for division of probabilities across the tails.
  • Examples demonstrate how to compute probabilities using known mean and standard deviation values.