Population parameters such as mu and sigma2 are fixed but often unknown. Sample statistics use observed data to estimate them. **Object** | **Status before sampling** | **Example** Population parameter | Fixed, usually unknown | mu, sigma2 Sample statistic / estimator | Random variable | X, S_n2 Estimate | Realised numerical value | x, s_n2 Divisor-n sample statistics X= 1n _i=1nX_i, S_n2= 1n _i=1n(X_i- X)2= 1n X_i2- X,2 x= 1n _i=1nx_i, s_n2= 1n _i=1n(x_i- x)2= 1n x_i2- x,2 An estimator is a rule applied before observing the sample and therefore has a sampling distribution. An estimate is the value obtained after applying that rule to one realised sample.

Population parameters such as \(\mu\) and \(\sigma^2\) are fixed but often unknown. Sample statistics use observed data to estimate them.

\[\overline X=\frac1n\sum_{i=1}^nX_i,\qquad S_n^2=\frac1n\sum_{i=1}^n(X_i-\overline X)^2=\frac1n\sum X_i^2-\overline X^{,2}\]

Random-variable form

\[\overline x=\frac1n\sum_{i=1}^nx_i,\qquad s_n^2=\frac1n\sum_{i=1}^n(x_i-\overline x)^2=\frac1n\sum x_i^2-\overline x^{,2}\]

Realised-data form

An estimator is a rule applied before observing the sample and therefore has a sampling distribution. An estimate is the value obtained after applying that rule to one realised sample.

Estimators and Estimates

Estimators and Estimates

Object	Status before sampling	Example
Population parameter	Fixed, usually unknown	\(\mu,\ \sigma^2\)
Sample statistic / estimator	Random variable	\(\overline X,\ S_n^2\)
Estimate	Realised numerical value	\(\overline x,\ s_n^2\)