Lecture Notes: Introduction to Normal Probability Distributions

Key Characteristics of Normal Probability Curves

• Shape: Bell-shaped curve
• Mean (μ): Central value at the peak of the curve
• Standard Deviation (σ): Measures spread around the mean
  • Affects the shape of the curve
  • Curve can be short and squat or tall and skinny depending on σ
• Total Area: Always equal to 1

Key Feature of Normal Curves

• 68-95-99.7 Rule:
  • 68% of the data falls within one standard deviation (±1σ) from the mean
  • This property holds regardless of the mean or standard deviation values

Example: Standard Deviation and the Normal Curve

1. Quantitative Variable X:
   • Mean (μ) = 10, Standard Deviation (σ) = 2
   • Normal curve centered at μ=10
   • Values one standard deviation away:
     • μ + σ = 12
     • μ - σ = 8
   • Probability (X between 8 and 12) = 0.68 or 68%

Example: Baby Birth Weights

• Distribution: Normal
• Mean (μ): 3500 grams
• Standard Deviation (σ): 600 grams
• Probability of Weight within One Standard Deviation: 68%
  • Range: 2900 grams to 4100 grams

Inflection Points

• Definition: Points where the curve changes from concave up to concave down or vice versa
• Location: At ±1σ from the mean
• Each normal curve has two inflection points corresponding to one standard deviation above and below the mean

Effect of Changing Standard Deviation

• Illustration with Applet:
  • Adjusting σ changes the spread and the shape of the curve
  • Despite changes in σ, the area within one standard deviation remains 68%

Conclusion

• Regardless of the shape (short and squat or tall and skinny), the area under the curve within one standard deviation of the mean is always 68%
• Inflection points occur at one standard deviation from the mean, influencing the curve's concavity.