Random Sample
Statistics & ProbabilityA random sample is a subset of a population in which every member has an equal chance of being selected.
Definition
A random sample means that everyone in a group has an equal and fair chance of being picked for the study. No one is chosen on purpose; it is like drawing names from a hat.
Example
A teacher puts every student's name on a slip of paper, mixes them up, and draws $10$ names to survey. That is a random sample.
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.
Definition
A simple random sample (SRS) is a sample drawn such that every possible sample of size $n$ has an equal probability of being selected. Random sampling reduces bias and ensures that sample statistics are valid estimators of population parameters.
Example
A school assigns each of its $800$ students a number from $1$ to $800$. A random number generator picks $80$ numbers. The $80$ corresponding students form a simple random sample for a study on study habits.
Key Insight
Other random sampling methods include stratified sampling (dividing into subgroups first), cluster sampling, and systematic sampling. Each has tradeoffs in cost, precision, and feasibility.
Definition
In a simple random sample of size $n$ from 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
Stratified random sampling partitions the population into $H$ strata and draws 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
Survey sampling theory (Cochran, 1977) extends these ideas to complex designs with unequal probabilities, requiring design weights in estimation to maintain unbiasedness.