A sample is a subset of a population selected for observation or measurement. Because observing an entire population is often impractical or impossible, statistics works with samples and uses them to estimate population parameters.
A random sample is one where each member of the population has a known probability of selection. Simple random sampling gives each member equal probability; stratified, cluster, and systematic sampling use more structured selection methods. The sampling method determines what inferences are valid: a biased sample (one that systematically over- or under-represents parts of the population) leads to biased estimates.
Sample statistics — the sample mean, sample variance, sample median — are calculated from the observed data and used to estimate the corresponding population parameters. The law of large numbers guarantees that sample statistics converge to population parameters as the sample size grows. The central limit theorem guarantees that the sampling distribution of the mean approaches a normal distribution, regardless of the population’s shape, for large enough samples.