Understanding Sample Means and Distributions

Jan 20, 2025

Lecture on Sample Means and Their Distributions

Introduction

  • Exploration of sample means.
  • Importance of understanding simulating and examining sample means.
  • Sample means featured frequently in exams, particularly in multiple-choice questions.

Simulating a Population

  • Example simulation: town of 600 people with varying heights.
  • Heights are approximately normally distributed.
  • A sample of 10 people is taken to find the sample mean.
    • Example sample mean: 144.5 cm.
    • Sample means vary across different samples.

Distribution of Sample Means

  • Multiple samples were taken (200 times).
  • Distribution of sample means is approximately normal.
  • Larger samples (e.g., 30 people) produce a more pronounced normal distribution.
  • Variability in sample means shows decreased range with larger samples.

Population Mean vs. Sample Mean

  • Actual mean of population (census) is 150 cm.
  • Average of sample means is close to the actual population mean.
  • Sample mean as a random variable changes depending on the sample.

Simulating Different Distributions

  • Sample means from non-normally distributed populations:
    • Uniform distribution (e.g., school grades).
    • Bimodal distribution (e.g., basketballers and wheelchair basketballers).
  • Even non-normal populations yield a normal distribution of sample means.

Properties of Sample Means

  • Approximate Normality: When sample size (n) is large, distribution of sample means is approximately normal.
  • Mean of Sample Means: Equal to the mean of the population.
  • Standard Deviation of Sample Means:
    • Smaller than the population's standard deviation.
    • Decreases as sample size increases.

Standard Deviation Calculation

  • Formula: Standard deviation of the population divided by the square root of the sample size.
  • Larger sample sizes reduce standard deviation of sample means.

Standard Normal Distribution of Sample Means

  • For large samples (n > 30), sample means approximate a standard normal distribution.
    • Mean = 0
    • Standard Deviation = 1

Conclusion

  • Understanding sample means is crucial for interpreting statistical data.
  • The lecture covers theoretical concepts and practical simulations.
  • Key takeaway: Sample means of populations form a normal distribution, facilitating analysis and prediction.