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.