A population parameter is often unknown, so we use sample data to estimate it. A point estimator is a statistic designed for that purpose.

The estimator is the rule; the estimate is the numerical value produced by one particular sample.

For example, the sample mean

$$\bar X=\frac1n\sum_{i=1}^n X_i$$

is an estimator of a population mean $\mu$. If one sample gives $\bar x=12.4$, then $12.4$ is the estimate.

Bias

An estimator $T$ of a parameter $\theta$ is unbiased when

$$E[T]=\theta.$$

Bias measures systematic displacement of the estimator's sampling distribution from the true parameter.

Variability

Two unbiased estimators can have different precision. An estimator with a smaller standard error varies less from sample to sample and therefore tends to give values closer to its target.

Good estimation often balances bias and variability rather than judging an estimator by one observed sample alone.

Consistency

An estimator is consistent when it approaches the true parameter as the sample size grows under the assumed sampling model.

For example, the sample mean is a consistent estimator of the population mean under standard conditions.

Common estimators

The sample proportion

$$\hat p=\frac{X}{n}$$

estimates a population proportion, while $\bar X$ estimates a population mean.

A point estimate gives one best numerical summary from the sample. It does not by itself show how uncertain that estimate is; interval estimation adds that information.