Statistical population

​θ as $n \to \infty$n→∞. Efficiency, another desirable trait, refers to an estimator having the smallest variance among unbiased alternatives, though it is often assessed relative to a benchmark like the Cramér-Rao lower bound. Interval estimation builds on point estimates by constructing confidence intervals that quantify uncertainty. For the population mean, a common normal approximation yields the interval $\bar{x} \pm z_{\alpha/2} \frac{s}{\sqrt{n}}$xˉ±zα/2​n

​s​, where $z_{\alpha/2}$zα/2​ is the critical value from the standard normal distribution for a $1-\alpha$1−α confidence level, and $n$n is the sample size; this interval captures $\mu$μ with probability $1-\alpha$1−α in repeated sampling. The Central Limit Theorem plays a crucial role here, stating that for large $n$n (typically $n \geq 30$n≥30), the sampling distribution of $\bar{x}$xˉ is approximately normal with mean $\mu$μ and standard error $\sigma / \sqrt{n}$σ/n

​ (estimated by $s / \sqrt{n}$s/n

​), even if the population distribution is non-normal, enabling these normal-based intervals. A practical example is estimating the national unemployment rate, a key population parameter representing the proportion of the labor force without jobs but seeking work. The U.S. Bureau of Labor Statistics conducts the Current Population Survey, a monthly sample of about 60,000 households, to compute point estimates like the sample proportion of unemployed individuals, yielding the national rate (e.g., 4.3% as of August 2025).^[22] Confidence intervals from this sample, such as $\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$p^​±zα/2​np^​(1−p^​)​

​, provide a margin of error—often about 0.2 to 0.3 percentage points at 90% confidence—indicating the precision of the estimate relative to the full population.

Applications in Statistics

In Descriptive Statistics

In Inferential Statistics