Why Samples Differ

Why your sample differs from mine

• Take two random samples from the same population — their results won't match exactly.
• A statistic 统计量 (like $\bar{x}$ or $\hat{p}$) varies from sample to sample.
• This built-in wobble is sampling variability 抽样变异性.
• It's not a mistake — it's the unavoidable nature of sampling.

Parameter vs. statistic

• A parameter 参数 is a fixed number describing the whole population (e.g. true mean $\mu$).
• A statistic is a number computed from a sample (e.g. sample mean $\bar{x}$).
• The parameter is unknown and constant; the statistic is known but varies.
• We use the varying statistic to estimate the fixed parameter.

Five random samples from one population give five different sample proportions

The sampling distribution

• Imagine taking every possible sample and recording the statistic each time.
• The distribution of all those values is the sampling distribution 抽样分布.
• It shows how the statistic behaves: its center, spread, and shape.
• This is the key idea that makes inference possible.

Variability means uncertainty

• Because the statistic varies, a single sample gives an estimate with uncertainty 不确定性.
• We can't be sure our one $\bar{x}$ equals the true $\mu$.
• But the sampling distribution tells us how far off we're likely to be.
• Quantifying that uncertainty is the whole point of Units 5–9.

Write from a statistic that varies

• Many random samples are drawn from one population.
• Each sample has its own sample mean.