Sampling Distributions: Mean, Proportion & CLT Guide

A sampling distribution is the probability distribution of a statistic, the pattern of values you would get if you drew many samples and computed, say, the sample mean each time. One sample gives one x̄; the sampling distribution describes how x̄ varies across all possible samples.

It matters because inference is built on it. When you estimate a population mean from one sample, the sampling distribution tells you how far that single estimate is likely to be from the truth. Confidence intervals and hypothesis tests are just clever readings of sampling distributions.

Key Concepts

• Sampling distribution of the mean: the distribution of X̄ across repeated samples. Its mean is μ and its standard deviation is σ/√n, called the standard error.
• Sampling distribution of a proportion: the distribution of p̂ across samples. Its mean is p and its standard error is √(p(1-p)/n).
• Centre: both distributions are centred at the true parameter (μ and p), which is why x̄ and p̂ are unbiased estimators.
• Spread shrinks with n: quadrupling the sample size halves the standard error, so bigger samples give steadier statistics.
• Shape via CLT: for large n, both sampling distributions are approximately normal, regardless of the population's shape.

Formulas

Mean of X̄: E[X̄] = μ; standard error of the mean: SE(X̄) = σ/√n; with s replacing σ when σ is unknown: SE ≈ s/√n. For a proportion: E[p̂] = p, SE(p̂) = √(p(1-p)/n). Normal approximations: X̄ ≈ N(μ, σ²/n) and p̂ ≈ N(p, p(1-p)/n) for large n. The proportion formulas need np ≥ 10 and n(1-p) ≥ 10 for the normal approximation to behave.

Worked Example

In a city, 60% of commuters use the metro (p = 0.6). A survey samples n = 100 commuters. The sampling distribution of p̂ has mean 0.6 and SE = √(0.6 × 0.4/100) = √(0.0024) ≈ 0.049. What is the probability the sample shows more than 70% metro users? Z = (0.70 - 0.60)/0.049 = 0.10/0.049 ≈ 2.04. P(Z > 2.04) = 1 - 0.9793 ≈ 0.0207, about 2.1%. So a sample result above 70% would be a fairly unusual fluke when the truth is 60%.

Common Misconceptions

• It is a distribution of statistics, not data: the sampling distribution of X̄ is not the histogram of your 100 measurements; it is the imagined histogram of millions of sample means. Confusing the two mixes up σ with σ/√n.
• Bigger n shrinks spread, not centre: increasing n does not move the distribution's mean; it was already μ. It only tightens the spread around it.
• Approximately normal needs conditions: for proportions, check np and n(1-p) are both at least 10. With p = 0.02 and n = 100, the normal approximation is unreliable.

Real-World Applications

• Poll margins of error: the familiar ±3% comes straight from SE(p̂) = √(p(1-p)/n) with n ≈ 1000.
• Quality auditing: deciding whether a sample mean weight of 498 g (target 500 g) is a real problem or ordinary sampling wobble.
• Clinical trials: judging whether the observed difference between treatment and control groups could just be sampling variation.

Sampling distributions describe how statistics wobble from sample to sample: centred on the truth, with spread σ/√n for means and √(p(1-p)/n) for proportions, and approximately normal for large n. Once this picture is clear, estimation and testing become natural next steps rather than magic formulas.