Sampling Error Calculator
Find the sampling error for a proportion survey at any confidence level. Enter the proportion estimate, sample size and confidence level to get the sampling error (margin of error) and see how it shrinks as sample size grows. Also shows the minimum n needed for ±3% and ±5% precision.
Confidence level
Maximum expected difference between sample proportion and true population proportion
- 1
p(1 − p)
0.5 × (1 − 0.5) = 0.25 - 2
Standard error SE
√(0.25 ÷ 400) = 0.025 - 3
Sampling error = z* × SE
1.96 × 0.025 = 0.0490Half-width of the confidence interval at the chosen level.
How does this calculator work?
Sampling error = z* × √(p(1−p)/n). Use p = 0.5 for the worst-case bound. For 95% confidence and p = 0.5, a sample of 385 gives ±5% precision; 1,067 gives ±3%. Sampling error shrinks with 1/√n — you must quadruple the sample to halve the error.
Formula
How this is calculated
Sampling error is the unavoidable discrepancy that arises when a sample — rather than a full census — is used to estimate a population characteristic. For a proportion (e.g. the share of voters who prefer a candidate), the sampling error is quantified by the margin of error: z* × √(p(1−p)/n), where p is the proportion estimate (use 0.5 if unknown to get the largest, most conservative error bound), n is the sample size, and z* is the critical value for the chosen confidence level (1.645 for 90%, 1.96 for 95%, 2.576 for 99%).
The curve in the output shows how sampling error decreases as n increases — notice the curve is steep for small n and flattens for large n. This is because sampling error scales with 1/√n: doubling your sample reduces error by about 29%, not 50%. To halve the error you must quadruple the sample size. This is why surveys of 1,000–1,500 respondents are standard for national polls — beyond about 1,500, the marginal gain in precision becomes very small.
This calculator assumes simple random sampling with replacement from a large population. If you are sampling more than 5% of the population, a finite population correction (FPC) reduces the error further. Also note that sampling error measures only random sampling variation — it does not capture non-sampling errors such as question wording, response bias or non-response.
Frequently asked questions
The expression p(1−p) is maximised when p = 0.5, giving the largest possible standard error for a given sample size. Using 0.5 as a conservative estimate ensures the calculated sampling error is never understated — regardless of what the true proportion turns out to be.
The standard error (SE = √(p(1−p)/n)) measures the spread of the sampling distribution in the same units as the proportion. The sampling error (or margin of error) is z* × SE — it converts the SE into a practical error bound at a specific confidence level. The sampling error is wider for higher confidence levels.
Not significantly, provided the population is large compared with the sample (the usual assumption). The sampling error formula only depends on n, not on population size N. A poll of 1,000 people has approximately the same ±3% margin of error whether the population is 1 million or 100 million.
Also known as
TG we-Calculate Editorial Team. (2026). Sampling Error Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/sampling-error-calculator
TG we-Calculate Editorial Team. "Sampling Error Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/sampling-error-calculator.
TG we-Calculate Editorial Team, "Sampling Error Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/sampling-error-calculator
@misc{wecalculate_sampling_error_calculator, title = {Sampling Error Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/sampling-error-calculator}}, year = {2026}, note = {TG we-Calculate} }
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