Random Sampling

Why randomize the choice?

• Random sampling 随机抽样 uses chance to decide who gets into the sample.
• Chance has no agenda, so it avoids the human tendencies that create bias.
• A random sample tends to look like the population in the long run.
• Randomness is what makes a fair generalization possible.

The simple random sample

• A simple random sample (SRS) 简单随机样本 gives every group of $n$ individuals an equal chance.
• Carry it out: number the population, then draw numbers (random digits or a hat).
• Every individual — and every combination — is equally likely.
• The SRS is the gold-standard idea the others build on.

Random outcomes: dice make each face equally likely under fair conditions

Stratified, cluster, systematic

• Stratified 分层: split into similar groups (strata), then take an SRS within each.
• Cluster 整群: split into groups (clusters), randomly pick whole clusters, sample everyone in them.
• Systematic 系统: pick a random start, then take every $k$-th individual.
• Stratify to guarantee representation of each group; cluster for convenience when groups are scattered.

Randomness enables generalization

• Because a random method has no bias, its results generalize to the population.
• The sample statistic becomes a trustworthy estimate of the population value.
• Non-random ("convenience") samples give you no such guarantee.
• Later units' inference formulas assume the sample was random.

Write from a stratified sample

• The population is split into year groups.
• A random sample is taken inside each year group.