Sampling Distributions
A sampling distribution is the distribution of a statistic — like a sample mean or sample proportion — across every possible sample of a given size, and it's what lets you judge how far one sample might land from the truth.
Definition
A statistic computed from a sample (a sample proportion p̂, a sample mean x̄) changes from sample to sample. The sampling distribution is the distribution of that statistic over all possible samples of size n. Its center, spread, and shape are what make inference possible: they tell you exactly how much sampling variability to expect from chance alone.
The formula & procedure
For a sample proportion: the mean of p̂ is p, and its standard deviation is √( p(1 − p) / n ) when the 10% condition holds; the shape is roughly Normal when np ≥ 10 and n(1 − p) ≥ 10. For a sample mean: the mean of x̄ is μ, and its standard deviation is σ/√n; the shape is Normal if the population is Normal, and approximately Normal for large n by the Central Limit Theorem (a common rule of thumb is n ≥ 30).
Worked example
Suppose 60% of a large streaming service's subscribers finished a certain series, so p = 0.60, and you look at samples of n = 100 subscribers. The sampling distribution of p̂ has mean 0.60 and standard deviation √(0.60·0.40/100) = √0.0024 ≈ 0.049. Because np = 60 and n(1 − p) = 40 are both ≥ 10, it's roughly Normal, so a typical sample proportion lands within about 0.60 ± 0.098 (two standard deviations) — roughly 0.50 to 0.70. A sample showing 0.52 is unremarkable; a sample showing 0.40 would be surprisingly far out.
The classic mistake
Confusing the spread of the population (or of one sample) with the spread of the sampling distribution. As n grows, the sampling distribution gets narrower — its standard deviation shrinks by a factor of √n — even though the population's own spread never changes. Students also misread the Central Limit Theorem as a claim that the raw data become Normal; it's about the distribution of the statistic, not the individual values.
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