Sampling Distributions & the Central Limit Theorem
Big Idea: When we take many random samples from any population and compute their means, those sample means form a predictable, bell-shaped distribution — regardless of the shape of the original population!
1. Key Definitions
Population vs. Sample
- Population: The entire group we want to study. Has parameters $\mu$ (mean) and $\sigma$ (standard deviation).
- Sample: A subset of size $n$ drawn from the population. Has statistics $\bar{x}$ (sample mean) and $s$ (sample standard deviation).
Sampling Distribution of $\bar{x}$
Imagine repeating this process thousands of times:
- Draw a random sample of size $n$ from the population
- Compute the sample mean $\bar{x}$
- Record it
The distribution of all possible sample means is called the sampling distribution of $\bar{x}$.
2. Properties of the Sampling Distribution
Three Fundamental Properties
| Property | Formula |
|---|---|
| Mean of $\bar{x}$ | $$\mu_{\bar{x}} = \mu$$ |
| Standard Deviation of $\bar{x}$ (Standard Error) | $$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$$ |
| Shape | Approximately Normal when $n$ is large enough |
The standard error $\frac{\sigma}{\sqrt{n}}$ decreases as $n$ increases — larger samples give more precise estimates!