Sampling Distributions

What Is a Sampling Distribution?

A sampling distribution is the distribution of a statistic (like $\bar{x}$ or $\hat{p}$) computed from all possible samples of a given size from a population.

Key Idea

Individual statistics vary from sample to sample. The sampling distribution describes this variability.

Sampling Distribution of the Sample Proportion $\hat{p}$

For a sample of size $n$ from a population with proportion $p$:

$\mu_{\hat{p}} = p$

$\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$

Conditions for approximate Normality:

1. Random: Data comes from a random sample or experiment
2. 10% Condition: $n < 0.10N$ (for independence)
3. Large Counts: $np \geq 10$ and $n(1-p) \geq 10$

When conditions are met: $\hat{p} \approx N\left(p, \sqrt{\frac{p(1-p)}{n}}\right)$

Sampling Distribution of the Sample Mean $\bar{x}$

For a sample of size $n$ from a population with mean $\mu$ and standard deviation $\sigma$:

$\mu_{\bar{x}} = \mu$

$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

$\sigma_{\bar{x}}$ is called the standard error of the mean.

Unbiased Estimators

A statistic is an unbiased estimator of a parameter if its sampling distribution is centered at the parameter value.

• $\bar{x}$ is unbiased for $\mu$
• $\hat{p}$ is unbiased for $p$
• $s^2$ is unbiased for $\sigma^2$
• $s$ is slightly biased for $\sigma$

Effect of Sample Size

As $n$ increases:

• The sampling distribution becomes less spread out (more precise)
• $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$ decreases
• The distribution becomes more Normal (CLT)

Variability vs. Bias

• Bias: Systematic error — is the center of the sampling distribution at the right place?
• Variability: Random error — how spread out is the sampling distribution?

AP Tip: The concept of sampling distributions is the foundation for inference. Understand that we're not talking about the distribution of the data, but the distribution of a statistic across many samples.