Sample (Statistics)
A sample is a subset of a population selected for study in order to draw conclusions about the whole population.
Definition
A sample is a smaller group chosen from a larger population to represent it: instead of studying everyone, a subset is selected and studied in order to make inferences about the whole. The quality of a sample depends on how it is selected, a random sample gives every member an equal chance of being chosen and tends to be unbiased. Formally, a sample $\{x_1, \ldots, x_n\}$ is an independent and identically distributed (i.i.d.) draw from population distribution $F(x; \theta)$, and sample statistics (e.g., $\bar{x}$, $s^2$) are functions of the sample used as estimators of population parameters, judged by properties like unbiasedness, consistency, and efficiency.
Example
To find out what snack the whole school prefers, you might ask $50$ randomly chosen students; those $50$ students are your sample. In a school of $1{,}200$ students, a researcher might randomly select $120$ students ($10\%$) and record their daily screen time, using the mean screen time of those $120$ as a sample statistic to estimate the population mean. The sample mean $\bar{x} = \frac{1}{n}\sum x_i$ is an unbiased estimator of $\mu$: $E[\bar{x}] = \mu$, while the sample variance $s^2 = \frac{1}{n-1}\sum (x_i - \bar{x})^2$ uses $n-1$ (Bessel's correction) to remain unbiased for $\sigma^2$.
Key Insight
A good sample is like a taste test: a small spoonful can tell you a lot about the whole pot, as long as you stir it first, or randomize it. Sample size matters too: larger samples give more precise estimates, but a large biased sample can be worse than a small random one. The central limit theorem guarantees that $\bar{x}$ is approximately normally distributed for large $n$ regardless of the population distribution, enabling confidence intervals and hypothesis tests.