Lecture on Standard Normal Distribution and Central Limit Theorem

Introduction

• Discusses standard normal distribution and central limit theorem.
• Use of data displays to assess and describe distributions and sample quality.
• Measures of central tendency can describe the shape of distributions.
  • Normal distribution: mean, median, mode are the same.
  • Skewed distribution: order of mean, median, mode indicates skewness.

Standard Normal Distribution

• Characteristics:
  • Bell-shaped curve.
  • Mean = 0, Standard deviation = 1.
• Conversion Process: Centering and Standardization.
  • Centering: Subtract mean from each data point.
  • Standardization: Divide centered data by the standard deviation, resulting in z-scores.

Example: Height Data

• Mean: 65.24, SD: 5.008.
• Z-score formula: (Observed value - Mean) / Standard deviation.
• Example Calculation: 63 inches height results in a z-score of -0.4473.

Importance of Z-scores

• Z-scores indicate how many standard deviations a value is from the mean.
• Related to area under the curve in normal distribution.
  • ±1 SD: 68.26% of data.
  • ±2 SD: 95.44%.
  • ±3 SD: 99.74%.

Applications and Examples

• Use z-scores to determine probabilities and areas under the curve.
  • Example: Student's test scores converted to z-scores to find proportions above/below certain scores.

Bognar Website for Calculations

• Tool for calculating areas under normal distribution curves.
• Input z-value or p-value to calculate the other.
• Provides graphical interface for different areas of distribution.

Options in Drop-Down Menu

• Greater than observed value.
• Less than observed value.
• Two-sided value.
• In between values.

Central Limit Theorem (CLT)

• Concept: As sample size increases, distribution of sample means approaches normal distribution.
• Population mean (µ) and standard deviation (σ).
• Sample mean distribution: approximately normal with mean = µ and standard deviation = σ/√n.

Example

• Data with non-normal distribution becomes more normal as sample size increases.
• Asymptotic behavior: Distribution approximates standard normal with large sample sizes.

Key Takeaways

• Standard normal distribution is a fundamental concept in statistics.
• Z-scores are essential for understanding data relative to the mean.
• Central Limit Theorem allows for approximation of distributions with large sample sizes.