Distribution of the Sample Mean
Understand how sample means behave and why they become normally distributed through the Central Limit Theorem.
Sampling Distribution
A statistic (like the sample mean ) is a random variable because its value varies from sample to sample.
Sampling Distribution
The probability distribution of all possible values of a statistic computed from a sample of size .
Key Insight: If you repeatedly take samples of size n from a population and calculate their means, the distribution of those sample means is the sampling distribution of .
Mean and Standard Error
Equals the population mean
Decreases as n increases
Important: As sample size increases, the standard error decreases. This means sample means cluster more tightly around μ with larger samples.
Normal Population Rule
If the underlying population is normally distributed, the sampling distribution of is also normal, regardless of the sample size .
This means even with small samples (n = 5, n = 10), you can use normal distribution methods to calculate probabilities about if the population is normal.
Central Limit Theorem (CLT)
Regardless of the shape of the underlying population, the sampling distribution of becomes approximately normal as the sample size increases.
Sample Size Requirements
Typically sufficient for most non-normal populations.
May require larger n (e.g., 50+).
Even for non-normal populations: The mean remains and the standard deviation remains .
Try It Yourself
Central Limit Theorem Simulator
Population Distribution
Sampling Distribution of
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