| Term | Meaning |
|---|---|
| Population | The complete collection of individuals or items under consideration. |
| Sample | A subset selected from the population for analysis. |
| Random sample | A sample selected by a stated random mechanism; in the usual H2 model the observations are independent and identically distributed. |
| Sample size | The number of observations, denoted by \(n\). |
Random sampling describes the selection design. It is not synonymous with an unbiased estimator, which is a separate sampling-distribution property.
Sample total and sample mean
| Metric | Sample total \(T_n=\sum_{i=1}^nX_i\) | Sample mean \(\overline X=\frac1n\sum_{i=1}^nX_i\) |
|---|---|---|
| Expectation | \(E(T_n)=n\mu\) | \(E(\overline X)=\mu\) |
| Variance | \(\operatorname{Var}(T_n)=n\sigma^2\) | \(\operatorname{Var}(\overline X)=\frac{\sigma^2}{n}\) |
| Standard deviation | \(\operatorname{SD}(T_n)=\sqrt n\,\sigma\) | \(\operatorname{SD}(\overline X)=\frac{\sigma}{\sqrt n}\) |
The standard deviation of the sample mean is its standard error: \(\operatorname{SE}(\overline X)=\dfrac{\sigma}{\sqrt n}\). A larger \(n\) reduces this spread and improves precision.
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