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Understanding Sample Means and Distributions
Jan 20, 2025
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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.
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