Why Samples Differ
Introduced| English |
|---|
| statistic/stəˈtɪstɪk/ |
| sampling variability/ˈsæmplɪŋ ˌveərɪəˈbɪlɪti/ |
| parameter/pəˈræmɪtə/ |
| sampling distribution/ˈsæmplɪŋ ˌdɪstrɪˈbjuːʃn/ |
| uncertainty/ʌnˈsɜːtənti/ |
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.
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.
Keep parameter and statistic straight — they use different symbols on purpose. A parameter ($\mu$, $p$, $\sigma$) is a fixed population truth we usually don't know; a statistic ($\bar{x}$, $\hat{p}$, $s$) comes from a sample and changes every time. The sampling distribution describes the statistic, not any one sample.
A population has true mean $\mu = 50$.
- Sample 1 gives $\bar{x} = 48$; Sample 2 gives $\bar{x} = 53$ — sampling variability.
- Neither equals $\mu$ exactly, and that's expected.
- The sampling distribution of $\bar{x}$ is centered at $50$ and shows the typical spread of these estimates.
A statistic varies from sample to sample (sampling variability), while a parameter is a fixed population value. The sampling distribution is the distribution of a statistic over all possible samples — it captures the uncertainty in using one sample to estimate a parameter.
A sampling distribution
The statistic varies from sample to sample around the true parameter.
Which is a parameter (not a statistic)?
μ is a fixed population value — a parameter. The others come from samples.
Two random samples from the same population will usually give slightly different statistics.
That's sampling variability — expected, not an error.
The distribution of a statistic over all possible samples is the ___ distribution (one word).
The sampling distribution describes how the statistic behaves.
A parameter is best described as...
Parameters (μ, p, σ) are fixed population truths.
Because a statistic varies, a single sample gives an estimate with some uncertainty.
The sampling distribution quantifies that uncertainty.