Normal Distribution (Bell-Shaped Curve) Lecture

Overview of the Curve

• Population Mean (µ) and Standard Deviation (σ):
  • µ (mean) is at the center
  • σ (standard deviation) marks intervals to the right and left of the mean
  • Extends from -∞ to ∞
• Area Under the Curve:
  • Represents probability of events between two points
• 68-95-99.7 Rule:
  • 68.268% within 1σ
  • 95.45% within 2σ
  • 99.73% within 3σ

Symmetry and Area Calculations

• The curve is symmetrical:
  • 68.268% divided by 2 = 34.134% on either side of the mean
  • The left side mirrors the right
• Subtracting percentages for different regions:
  • (95.45% - 68.268%) / 2 = 13.591% between 1σ and 2σ
  • (99.73% - 95.45%) / 2 = 2.14% between 2σ and 3σ
  • (100% - 99.73%) / 2 = 0.135% in the tails beyond 3σ

Important Considerations

• Total probability under curve: 100% or 1
• Each side of the mean: 50% or 0.5

Practical Examples

Problem 1: Physics Class Test Scores

• Given: Mean = 75, σ = 7, 800 students

• Part A: Percentage between 68 and 82

  • Calculate probability:
    • 34.134% (mean to +1σ) + 34.134% (mean to -1σ)
    • Total: 68.268%
  • Verify with definite integral:
    • Result: approximately 68.269%

• Part B: Number of students between 61 and 89

  • Probability between two standard deviations (61 to 89):
    • 95.45%
    • Calculation:
      • 13.591% + 34.134% + 34.134% + 13.591%
    • Verify with definite integral:
      • Result: ~95.45%
  • Number of students:
    • 800 * 0.9545 = ~764 students

• Part C: Probability of score between 54 and 75

  • Probability:
    • 49.865%
    • Calculation:
      • 2.14% + 13.591% + 34.134%
    • Verify with definite integral:
      • Result: ~49.865%

• Part D: Probability of score ≥ 96

  • Percentage beyond 3σ (96 and above):
    • 0.135%
    • Number of students:
      • 800 * 0.00135 = 1 student
    • Verify with definite integral from 96 to ∞ (use very high number to approximate ∞):
      • Result: 0.135%

Using Online Calculators

• Wolfram Alpha:
  • Can be used to calculate definite integrals
  • Input function and limits to obtain probability
  • Example: ∫_{68}^{82} e^{- (x - 75)^2 / (2 * 7^2)} / (7√(2π)) dx

Summary

• Understanding the normal distribution and area under the curve is essential for solving probability problems
• Practical applications include calculating percentages and number of occurrences within specified ranges of standard deviations