Sampling Distribution of x̄ and the Central Limit Theorem

Sampling Distributions and the Central Limit Theorem

By the end of this lesson, you will be able to describe the sampling distribution of a statistic, apply the Central Limit Theorem, calculate probabilities using sampling distributions of $\bar{x}$ and $\hat{p}$.

Parameters vs. Statistics

Parameter: a number that describes the population (unknown; what we're estimating).

• $\mu$ = population mean
• $\sigma$ = population standard deviation
• $p$ = population proportion

Statistic: a number computed from sample data (known; used to estimate the parameter).

• $\bar{x}$ = sample mean
• $s$ = sample standard deviation
• $\hat{p}$ = sample proportion

Sampling Variability

Each time you take a sample, you get a different sample mean or proportion. This sampling variability is expected and can be described mathematically.

Sampling distribution: the distribution of a statistic over all possible samples of size $n$ from the population.

Key properties:

• Center: is the statistic an unbiased estimator? (Is the mean of the sampling distribution equal to the parameter?)
• Spread: how much variability is there? (Standard error)
• Shape: is the distribution approximately normal?

Sampling Distribution of $\bar{x}$ (Sample Mean)

If we sample from a Normal population or if $n$ is large enough:

Mean of $\bar{x}$: $\mu_{\bar{x}} = \mu$

Standard deviation of $\bar{x}$ (standard error): $\sigma_{\bar{x}} = \dfrac{\sigma}{\sqrt{n}}$

$\bar{x}$ is an unbiased estimator of $\mu$ — the average of all possible sample means equals the population mean.

The spread decreases as $n$ increases: larger samples give more precise estimates.

The Central Limit Theorem (CLT)

The CLT is one of the most important theorems in statistics.

Statement: If $n$ is large (generally $n \geq 30$ is a common rule of thumb), the sampling distribution of $\bar{x}$ is approximately normal, regardless of the shape of the population distribution.