Introduction to Inference

Sampling Distributions

Understanding how sample statistics vary from sample to sample.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

How Much a Sample Result Bounces Around

A fair coin has a true probability of heads p = 0.5. Flip it 20 times and compute the sample proportion of heads, p̂ — it won't necessarily land exactly on 0.5. Before reading on: if you repeated this 20-flip experiment many times, would you expect p̂ to bounce around 0.5 by a little, or by a lot?

Definition — Sampling Distribution and the Role of Sample Size

A sampling distribution describes how a sample statistic, like p̂, varies across many different random samples of the same size. Its center is always the true population value (here, p = 0.5) — a sample statistic doesn't systematically favor being too high or too low. Its spread, σ_p̂ = √(p(1−p)/n), shrinks as the sample size n grows: larger samples produce sample statistics that cluster more tightly around the true value.

\sigma_{\hat p} = \sqrt{\frac{p(1-p)}{n}}

Worked Example — Simulated Results from Small Samples

Five separate simulated experiments of 20 coin flips each give sample proportions: 0.55, 0.40, 0.60, 0.45, 0.50 — all reasonably close to 0.5, but genuinely bouncing around rather than landing exactly on it every time. The formula confirms how much bounce to expect: σ_p̂ = √(0.5×0.5/20) = √0.0125 ≈ 0.112, so results within roughly 0.10 to 0.22 of 0.5 (within 1 to 2 standard deviations) are entirely ordinary — exactly the range these five simulated results fall in.

Worked Example — The Effect of a Larger Sample Size

Now compare n = 100 flips instead of 20: σ_p̂ = √(0.5×0.5/100) = √0.0025 = 0.05 exactly — less than half the spread of the n=20 case. A sample of 100 flips would typically produce a p̂ within about 0.05 of 0.5, a noticeably tighter cluster than the n=20 samples above. Larger samples don't change where the sampling distribution is centered — only how tightly it clusters there.

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

A single sample result that lands somewhat far from the true value isn't automatically evidence something is wrong — some spread is expected, and how much is 'expected' depends directly on the sample size.

Common Mistakes

Assuming a larger sample size changes the center of the sampling distribution.
The sampling distribution's center is always the true population value, regardless of sample size — only the spread shrinks as n grows, not the center itself.

Key Takeaways

A sampling distribution describes how a sample statistic varies across many possible samples of a given size.
Its center is the true population value; its spread shrinks as sample size grows, following σ_p̂ = √(p(1−p)/n).
Some sample-to-sample variability is always expected — the sampling distribution tells you how much is ordinary.

Summary

Understanding how p̂ varies from sample to sample is exactly what's needed to estimate an unknown population proportion with a stated margin of error, the final topic of this course.

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