Sampling Distribution of the Sample Proportion
Given a sample proportion p̂ and sample size n, find the standard error of p̂, the margin of error and the confidence interval at a chosen confidence level. The bell curve shows the sampling distribution centred at p̂ with the CI shaded.
Confidence level
Half-width of the confidence interval around the sample proportion
- 1
p̂(1 − p̂)
0.45 × (1 − 0.45) = 0.2475 - 2
Standard error SE
√(0.2475 ÷ 200) = 0.035178Spread of the sampling distribution of p̂. - 3
Margin of error ME = z* × SE
1.96 × 0.035178 = 0.0689
How does this calculator work?
The sampling distribution of p̂ is approximately normal with standard error SE = √(p̂(1−p̂)/n). The confidence interval is p̂ ± z*·SE, where z* = 1.96 for 95% confidence. This requires np̂ ≥ 10 and n(1−p̂) ≥ 10 for the normal approximation to hold.
Formula
How this is calculated
When a random sample of size n is drawn from a large population and a proportion p̂ is observed (for example, the fraction of respondents who prefer option A), the sample proportion p̂ is itself a random variable with a sampling distribution. By the Central Limit Theorem, for sufficiently large n this distribution is approximately normal with mean equal to the true population proportion p and standard deviation — called the standard error — of SE = √(p̂(1−p̂)/n). Because the true p is unknown, the plug-in estimate p̂ is used in the formula.
The margin of error is computed as z* × SE, where z* is the critical value from the standard normal distribution for the chosen confidence level: z* = 1.645 for 90%, 1.96 for 95% and 2.576 for 99%. The resulting confidence interval [p̂ − ME, p̂ + ME] is interpreted as: if the sampling procedure were repeated many times, the interval would contain the true proportion p in the stated percentage of repetitions.
The normal approximation is reliable when both np̂ and n(1−p̂) are at least 10 (some texts use 5). The calculator flags a warning when this condition is not met. For small samples or proportions near 0 or 1, exact binomial-based methods (e.g. the Clopper–Pearson interval) are more accurate.
Frequently asked questions
The standard error (SE) measures how much the sample proportion p̂ varies from sample to sample. It equals √(p̂(1−p̂)/n). A larger sample reduces SE proportionally to √n — so quadrupling the sample size halves the SE.
Use exact binomial methods (e.g. Clopper–Pearson or Wilson interval) when np̂ or n(1−p̂) is less than 10, or when p̂ is very close to 0 or 1. The Wilson interval is often preferred in practice as it has better coverage for small samples.
Not quite. The confidence level is a property of the procedure, not the specific interval. A 95% CI means that if you were to repeat the sampling process many times, 95% of the intervals constructed this way would contain the true proportion — but any individual interval either does or does not contain it.
Also known as
TG we-Calculate Editorial Team. (2026). Sampling Distribution of the Sample Proportion [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/sampling-distribution-sample-proportion-calculator
TG we-Calculate Editorial Team. "Sampling Distribution of the Sample Proportion." TG we-Calculate. 2026. https://we-calculate.com/calculator/sampling-distribution-sample-proportion-calculator.
TG we-Calculate Editorial Team, "Sampling Distribution of the Sample Proportion," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/sampling-distribution-sample-proportion-calculator
@misc{wecalculate_sampling_distribution_sample_proportion_calculator, title = {Sampling Distribution of the Sample Proportion}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/sampling-distribution-sample-proportion-calculator}}, year = {2026}, note = {TG we-Calculate} }
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