Unit 5: Sampling Distributions

Sampling Distribution of a Proportion

Describing the center, spread, and shape of the sampling distribution of p̂.

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

How Much a Sample Proportion Varies from Sample to Sample

Suppose 60% of all adults truly support a policy. A single random sample of 100 adults won't necessarily show exactly 60% support — sampling variability means p̂ bounces around from sample to sample. Before reading on: if you imagined taking thousands of different random samples of 100 adults and computing p̂ for each one, what shape do you think that collection of p̂ values would take?

Definition — Sampling Distribution of a Sample Proportion

The sampling distribution of p̂ describes how p̂ varies across all possible random samples of a given size. Its center is the true population proportion p (making p̂ an unbiased estimator — it doesn't systematically over- or under-shoot on average). Its spread is σ_p̂ = √(p(1−p)/n). Its shape is approximately Normal when both np ≥ 10 and n(1−p) ≥ 10 (the large counts condition).

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

Worked Example — Describing and Using a Sampling Distribution

For p = 0.60, n = 100: center = 0.60. Spread: σ_p̂ = √(0.6×0.4/100) = √0.0024 ≈ 0.049. Large counts check: np = 60 ≥ 10 and n(1−p) = 40 ≥ 10, both satisfied, so p̂ is approximately Normal(0.60, 0.049). Find P(p̂ ≥ 0.65): z = (0.65−0.60)/0.049 ≈ 1.02, and P(Z ≥ 1.02) ≈ 0.154. There's roughly a 15% chance that a random sample of 100 adults would show 65% or more support, purely from sampling variability, even though the true population support is only 60% — a single sample result can look meaningfully different from the truth just by chance.

A sample proportion is secretly a special case of a sample mean: coding each person's response as 1 (supports) or 0 (doesn't) turns p̂ into the mean of those 0s and 1s. That connection is why simulating many random samples of a binary outcome (like a coin flip or a weighted win/lose trial) and tracking the running proportion demonstrates exactly the same sampling-variability behavior described here.

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 large counts condition depends on the sample size and the value of p together — a small p (a rare outcome) needs a correspondingly larger n before the Normal approximation becomes trustworthy.

Common Mistakes

Using p̂ from a single sample in the σ_p̂ formula instead of the true population value p, when p is actually given.
When the true population proportion p is stated (as it was here, 0.60), the sampling distribution's formulas use that known p — a sample's own p̂ only substitutes in when p itself is unknown and being estimated.

Key Takeaways

The sampling distribution of p̂ is centered at the true population proportion p, with spread σ_p̂ = √(p(1−p)/n).
The large counts condition (np ≥ 10 and n(1−p) ≥ 10) justifies treating the sampling distribution as approximately Normal.
A sample proportion is a special case of a sample mean — the mean of a variable coded as 0s and 1s.

Summary

The next lesson extends this same idea to the sampling distribution of a sample mean, and introduces the theorem that makes it work even when the underlying population isn't Normal at all.

Track your progress

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