Random Sample
A random sample is a subset of a population in which every member has an equal chance of being selected.
Definition
A random sample means everyone in a group has an equal and fair chance of being picked for a study, like drawing names from a hat. Formally, a simple random sample (SRS) is a sample drawn such that every possible sample of size $n$ has an equal probability of being selected, which reduces bias and ensures that sample statistics are valid estimators of population parameters. In a finite population of size $N$, each subset of size $n$ has probability $1/\binom{N}{n}$ of being the sample; under i.i.d. assumptions, sample statistics are unbiased estimators and the central limit theorem applies, justifying normal-based inference.
Example
A teacher who puts every student's name on a slip of paper, mixes them up, and draws $10$ names to survey is using a random sample. A school might assign each of its $800$ students a number from $1$ to $800$ and use a random number generator to pick $80$ numbers, forming a simple random sample for a study on study habits. Stratified random sampling partitions the population into $H$ strata and draws an SRS of size $n_h$ from each; the stratified estimator has smaller variance than SRS when within-stratum variance is small relative to between-stratum variance.
Key Insight
Random selection is the best way to avoid bias: when selection is random, the sample is likely to reflect the whole population fairly. Other methods include stratified sampling (dividing into subgroups first), cluster sampling, and systematic sampling, each with tradeoffs in cost, precision, and feasibility. Survey sampling theory (Cochran, 1977) extends these ideas to complex designs with unequal probabilities, requiring design weights in estimation to maintain unbiasedness.