Statistic (Value)
Statistics & ProbabilityA statistic is a numerical value calculated from a sample that is used to estimate a population parameter.
Definition
A statistic is a number you calculate from a sample. It summarizes something about that sample, like the average or the most common value.
Example
You ask $20$ friends how many hours they sleep. The average of those $20$ answers is a statistic.
Key Insight
Statistics (the numbers) are what we can actually measure. They are our window into the larger population we cannot fully see.
Definition
A statistic is any numerical measure calculated from sample data, used to estimate or test claims about a population parameter. Common statistics include the sample mean ($\bar{x}$), sample standard deviation ($s$), and sample proportion ($\hat{p}$).
Example
A pollster surveys $1{,}000$ voters and finds $48\%$ support a bill. That $48\%$ is a sample statistic ($\hat{p}$). It estimates the true population proportion ($p$), which is the parameter.
Key Insight
Statistics are random variables: each new sample produces a slightly different value. The distribution of all possible values of a statistic is its sampling distribution.
Definition
A statistic $T(X_1, \ldots, X_n)$ is a measurable function of the sample random variables. A sufficient statistic contains all the information in the sample about $\theta$. The Rao-Blackwell theorem states that conditioning an unbiased estimator on a sufficient statistic yields an improved (or equal) estimator.
Example
For exponential data, the sample mean $\bar{x}$ is a sufficient statistic for the rate parameter $\lambda$. Any other estimator can be improved by replacing it with its conditional expectation given $\bar{x}$.
Key Insight
Data reduction via sufficient statistics is fundamental to efficient estimation: you lose no information about $\theta$ by summarizing the data through a sufficient statistic.