A sample changes each time.

The population proportion p is fixed but usually unknown. The sample proportion p̂ = X/n varies across random samples. Under independent Bernoulli sampling, its mean is p and its standard deviation is $\sqrt{\frac{p(1-p)}n}$ .

For sufficiently large samples, the sampling distribution is approximately normal. This approximation needs enough expected successes and failures. Increasing n reduces random sampling variation, but does not repair systematic bias such as a voluntary response sample.

INTERACTIVE MODEL

Try it. Watch it change.

0Population p (teal) · sample estimate (blue)1

Proportion p / initial estimate0.4Sample size n200Confidence level

p̂ = 0.4 · interval (0.3321, 0.4679)
Large-sample count check met

Each simulated sample uses independent Bernoulli trials. The population p stays fixed while p̂ varies. Before a draw, p is used as the illustrative estimate. Bounds outside [0, 1] are clipped only in the picture.

Explore: Simulate several random samples from the same population proportion. Compare their estimates as n changes.

Mean sample proportion

E(\hat p)=p

Standard deviation

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

WORKED EXAMPLE

With population p = 0.4 and n = 100, find the standard deviation of p̂.

$\operatorname{SD}(\hat p)$ = $\sqrt{\frac{0.4(0.6)}{100}}$ .
SD ≈ 0.04899.

Assumed knowledge

Binomial models, proportions and standard deviation.

Learning checkpoints & source

YOUR LEARNING CHECKPOINT

Recognise selection bias and random sampling.
Describe the sampling distribution of a proportion.

QCAA Mathematical Methods 2025 v1.3 · p. 30 ↗

READY TO TRY IT?

A sample changes each time.

With population p = 0.4 and n = 100, find the standard deviation of p̂.

Put the idea to work.