Normal Distribution Lecture Notes

Introduction

• Presenter: Mark from Ace Tutors
• Topic: Normal Distribution, a common probability distribution in statistics.
• Structure:
  • High-level overview
  • Usage in finding probabilities
  • Example demonstration

Normal Distribution Overview

• Appearance: Symmetric bell-shaped curve (Bell Curve).
• Center: Distribution’s mean.
• Spread: Decreases in probability as you move away from the mean.
• Significance: Most probability values are within three standard deviations of the mean.
  • 1 standard deviation: 68% of data
  • 2 standard deviations: 95% of data
  • 3 standard deviations: 99.7% of data

Importance

• Common in real-world applications.
  • Examples: Heights, weights, IQ scores follow a normal distribution.

Standardization

• Problem: Different applications have different means, standard deviations, and units.
• Solution: Standardize data to eliminate units:
  • Transforms distribution to have a mean of 0 and standard deviation of 1.
  • Allows comparison across different scales.

Z-Score

• Definition: Measures how many standard deviations a data point is from the mean.
• Formula: ( Z = \frac{(X - \mu)}{\sigma} )
  • Example: Height of 67 inches, with mean of 66 and standard deviation of 2:
    • Z-Score: ( \frac{(67 - 66)}{2} = 0.5 )
    • Interpretation: 0.5 standard deviation above the mean.

Calculating Probabilities

• Method: Use probability charts (Z-Charts) instead of formulas.
  • Z-Charts: Shows z-scores and associated probabilities.

Example of Using Z-Charts

• Z-Score 0.63:
  • Find in chart: Row 0.6, Column 0.03 -> Probability 0.7357
  • Interpretation: 73.57% chance of Z-Score less than 0.63.
• Chart Types:
  • Negative Z-Scores (left of mean)
  • Positive Z-Scores (right of mean)

Example: Pizza Size Probability

• Scenario: Pizza shop claims large pizzas are at least 16 inches.
  • Mean: 16.3 inches, Standard Deviation: 0.2 inches.
  • Questions:
    • Probability of a pizza less than 16 inches?
    • Probability of a pizza more than 16.5 inches?
    • Probability of a pizza between 15.95 inches and 16.63 inches?

Solution Steps

1. Draw Distribution: Centered at 16.3 inches.
2. Standardize: Use Z-Score for different measures.
   • 16 inches: Z = ( \frac{(16 - 16.3)}{0.2} = -1.5 )
   • 16.5 inches: Z = ( \frac{(16.5 - 16.3)}{0.2} = 1 )
   • 15.95 inches: Z = ( \frac{(15.95 - 16.3)}{0.2} = -1.75 )
   • 16.63 inches: Z = ( \frac{(16.63 - 16.3)}{0.2} = 1.65 )
3. Find Probabilities:
   • P(X < 16): 6.68% chance
   • P(X > 16.5): 15.87% chance
   • P(15.95 < X < 16.63): 91.04% chance

Conclusion

• Normal distribution is essential in statistics for real-world applications.
• Standardizing helps in comparing across different magnitudes and units.
• Z-Scores and Z-Charts are critical tools in calculating probabilities.