Find the minimum sample size needed for surveys, polls, and research studies. Enter your desired margin of error, confidence level, and population size to get the required sample count instantly.
The sample size is the number of observations or individuals included in a study. Too small a sample produces unreliable results with wide confidence intervals and high margins of error. Too large a sample wastes resources. Finding the right balance is essential for good research design.
The primary driver of sample size is the desired margin of error β how close you want your estimate to be to the true population value. The secondary driver is the confidence level β how certain you want to be that your interval contains the truth.
The standard formula for proportions is: n = ZΒ² Γ p Γ (1βp) / EΒ²
For example, at 95% confidence, 5% MOE, p = 0.5: n = 1.96Β² Γ 0.5 Γ 0.5 / 0.05Β² = 3.8416 Γ 0.25 / 0.0025 = 384.16 β 385
When your population is finite (known size N), the required sample is smaller than the infinite formula suggests. The finite population correction adjusts for this: n_adj = n / (1 + (nβ1)/N)
The FPC has a meaningful effect when the sample is more than about 5% of the population. For example, surveying a company of 200 employees requires fewer respondents than the 385 that the infinite formula suggests.
US national surveys conducted by organisations like Gallup, Pew Research Center, and polling agencies typically use samples of 1,000 to 1,500 respondents. At n = 1,000 with 95% confidence, the margin of error is approximately Β±3.1%. The US population (~335 million) is so large that the finite population correction is negligible.
The US Census Bureau runs the American Community Survey (ACS) with approximately 3.5 million households, giving extremely precise estimates for small geographic areas. The Current Population Survey (CPS) samples around 60,000 households monthly.
UK opinion polling for Westminster voting intention typically uses samples of 1,000 to 2,000 adults, weighted to be representative of the UK population (~67 million). With n = 1,000, MOE β Β±3.1%. The Office for National Statistics (ONS) uses much larger samples for its labour market surveys β for example, the Labour Force Survey (LFS) interviews approximately 40,000 households per quarter.
NHS patient satisfaction surveys, the British Social Attitudes survey, and public health monitoring all use rigorous sample size calculations. The UK Statistics Authority regulates official statistics, requiring disclosure of sampling methodology.
Clinical trials require careful power analysis alongside sample size calculation. Power (1βΞ²) is the probability of detecting a real effect if one exists. Most trials target 80% or 90% power. A Phase III clinical trial testing a new drug might require thousands of participants to detect a clinically meaningful difference in outcome rates.
CONSORT guidelines require reporting of how sample size was determined in published trials. The US FDA and UK MHRA (Medicines and Healthcare products Regulatory Agency) both require pre-specified sample size calculations in clinical trial protocols.
In digital product development, A/B tests compare user behaviour between two versions. The minimum detectable effect (MDE) is the smallest change worth detecting. For a website converting at 4% baseline wanting to detect a 0.5% absolute improvement:
Online A/B testing tools (Optimizely, VWO, Google Optimize) all use sample size calculators internally based on the same statistical principles.
| Margin of Error | 90% CI Sample | 95% CI Sample | 99% CI Sample |
|---|---|---|---|
| 1% | 6,765 | 9,604 | 16,590 |
| 2% | 1,691 | 2,401 | 4,148 |
| 3% | 752 | 1,068 | 1,844 |
| 5% | 271 | 385 | 664 |
| 10% | 68 | 97 | 166 |
At 95% confidence with p = 50%, you need n = 385 for an infinite population. For a finite population of N = 1,000, the FPC reduces this to about 278. For N = 500, about 218.
p = 0.5 (50%) maximises pΓ(1-p) = 0.25, giving the largest (most conservative) sample size estimate. If you know the proportion will be far from 50% (e.g., 10%), you can use that value to get a smaller required sample.
When surveying a known finite population, the adjusted sample size is n_adj = n / (1 + (n-1)/N). It reduces the required sample when n is a significant fraction of N. For n = 385 and N = 1,000: n_adj = 385 / (1 + 384/1000) β 278.
Higher confidence requires larger samples. Going from 90% to 95% increases the required sample by about 42%. Going from 95% to 99% increases it by about 73%. The sample increases because higher confidence requires a larger z critical value.
UK political polls typically use 1,000β2,000 adults, giving a MOE of about Β±2.2β3.1% at 95% confidence. Polls are weighted to match the UK population profile and must meet British Polling Council standards for publication.
Inflate the required sample by the reciprocal of your expected response rate. If you need n = 385 and expect a 60% response rate, invite 385 / 0.60 = 642 people to participate.
Sample size calculation for surveys focuses on estimation precision (MOE). Power analysis for hypothesis tests focuses on the probability (power, 1-Ξ²) of detecting a real effect of specified size. Both methods produce required sample sizes but for different purposes.
Practically, yes β but with diminishing returns. The MOE is proportional to 1/βn, so quadrupling the sample halves the MOE. Beyond around n = 1,000β2,000 for national surveys, the statistical benefit of additional respondents is marginal relative to cost.
Results are for educational purposes. This calculator uses standard proportion-based formulas. For clinical trials, regulatory submissions, or complex research designs, consult a qualified statistician.