Unit 5: Sampling Distributions

Sampling Distribution of a Mean

Describing the center, spread, and shape of the sampling distribution of x̄, including the Central Limit Theorem.

Advanced25 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Sampling Distribution of a Proportion

Why Averages Behave More Predictably Than Individuals

Individual delivery times are right-skewed — most deliveries arrive close to on time, but a few take much longer, and none can be negative. Before reading on: if you averaged 64 delivery times at once, would you expect that average to be just as unpredictable and skewed as a single delivery time — or something different?

Definition — Sampling Distribution of a Sample Mean and the Central Limit Theorem

The sampling distribution of x̄ is centered at the population mean μ, with spread σ_x̄ = σ/√n. The Central Limit Theorem (CLT) states that for a large enough sample size, the sampling distribution of x̄ is approximately Normal — regardless of the shape of the original population. Averaging many observations smooths out individual extremes: a single unusually long delivery barely moves an average of 64 deliveries, while it would completely dominate a sample of 1.

\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}

Worked Example — Applying the Central Limit Theorem

A population of delivery times has μ = 30 minutes and σ = 8 minutes, with a known right-skewed shape. For random samples of n = 64: center = 30 minutes. Spread: σ_x̄ = 8/√64 = 8/8 = 1 minute. Even though individual delivery times are skewed, n = 64 is large enough for the CLT to apply, so x̄ is approximately Normal(30, 1) — a far tighter, more symmetric distribution than the individual deliveries themselves. Find P(x̄ > 32): z = (32−30)/1 = 2, and P(Z > 2) ≈ 0.0228. It would be fairly unusual — about a 2.3% chance — for a sample of 64 deliveries to average more than 32 minutes, if the true population mean really is 30.

Probability Simulator

Experiment

Type

Sides

Number of dice

Run a trial to see the outcome here.

Simulation

Target trials

Trials per tick

Track outcome

0 / 200 trials

Histogram — experimental vs theoretical

Running probability — Law of Large Numbers

Statistics

Mean

— experimental

3.5 theoretical

Variance

— experimental

2.917 theoretical

Standard deviation

— experimental

1.708 theoretical

Trials

0 experimental

— theoretical

Frequency table

Outcome	Experimental	Theoretical
1	—	0.167
2	—	0.167
3	—	0.167
4	—	0.167
5	—	0.167
6	—	0.167

Central Limit Theorem

Draws 300 independent samples of 5trials each from the current experiment, averages each sample, and histograms the resulting sample means — the distribution should look progressively more bell-shaped (normal) as the sample size grows, even though the experiment’s own distribution usually isn’t.

Sample size

Number of samples

Tip

The CLT's power is exactly that it doesn't require knowing or assuming the population's shape — only that the sample is large enough (a common rule of thumb is n ≥ 30, though a strongly skewed population may need more).

Common Mistakes

Believing the Central Limit Theorem means individual data values become more Normal as sample size grows.
The CLT is a statement about the sampling distribution of x̄ (the average across samples), not about the individual population values themselves — those keep whatever shape they started with, no matter how many samples are taken.
Confusing σ (the population standard deviation) with σ_x̄ (the standard deviation of the sample mean).
σ_x̄ = σ/√n is always smaller than σ itself (for n > 1) — averages vary less than individual values do, which is precisely the reason larger samples give more precise estimates.

Key Takeaways

The sampling distribution of x̄ is centered at μ, with spread σ_x̄ = σ/√n — shrinking as sample size grows.
The Central Limit Theorem guarantees x̄'s sampling distribution is approximately Normal for a large enough n, regardless of the population's own shape.
Averaging smooths out individual extremes, which is why x̄ varies far less from sample to sample than a single observation does.

Summary

This closes Unit 5: sampling distributions describe how statistics vary before any data is even collected. The remaining units build directly on this — using a sampling distribution to estimate and test claims about an unknown population parameter, starting with proportions.

Track your progress

← Sampling Distribution of a Proportion Back to Unit 5: Sampling Distributions