Why use this calculator
Sample size is the decision that fixes whether a survey can answer its question, and it is made before a single response arrives. Too few responses and the margin of error swallows the difference you were trying to detect, so the study ends with nothing that can be defended.
Too many and you have spent weeks of fieldwork and a chunk of budget buying precision nobody needed. The relationship is not linear, which is what surprises people: halving the margin of error quadruples the sample. This calculator makes that trade visible, applies the finite population correction so small groups are not over-sampled, and grosses the target up by your expected response rate so the number you take away is how many people to invite rather than how many you wish would answer.
How to use it
- Pick a confidence level first, with 95% being the working standard in almost every published survey; raise it to 99% only when the decision is safety-critical or clinical.
- Set the margin of error to the largest error you could defend in a report, remembering that ±5% on a 52% result means the true figure could be anywhere from 47% to 57%.
- Enter your best guess at the expected split, using 50% when you genuinely do not know, because that is the value that demands the largest sample and is therefore the safe assumption.
- Enter the population size if you know it, or zero if the group is effectively unlimited.
- Finally enter the response rate you have seen on similar work, so the invitation count reflects reality.
What each result means
- Required sample size
- Completed, usable responses needed to hit your margin of error at the chosen confidence.
- Invitations to send
- How many people to approach so that non-response still leaves you the required sample.
- Sample if the population were unlimited
- The uncorrected figure, which is what you need when the group is very large.
- Sample as a share of the population
- What fraction of the whole group you must reach; above about 10% the correction matters.
- Confidence level
- How often an interval built this way would contain the true population value over repeated surveys.
- Margin of error
- The half-width of the confidence interval around your result, expressed in percentage points either side.
- Assumed split
- The expected yes and no percentages used in the variance term of the formula.
- Z-score used
- The standard normal critical value matching your confidence level, taken from the normal distribution.
The formula
For an unlimited population
n₀ = z² × p(1 − p) / e²
Because p(1 − p) is largest at
p = 0.5, that value gives the most conservative sample
For a finite population of N the correction
n = n₀ / (1 + (n₀ − 1) / N)
The invitation count is n divided by the expected response rate. Sample size scales with 1 / e², so precision improves only with the square root of effort.
How this number is used in practice
National polling is where the arithmetic is most visible. A poll of around 1,000 adults carries a margin of error near ±3.1 percentage points at 95% confidence, whatever the size of the country, because once the population runs into millions the finite population correction stops making any difference.
That is why a sample of 1,000 works as well for a nation of 70 million as for one of 25 million, and why a headline claiming a two-point lead from such a poll is describing a difference smaller than its own error bar.
Inside an organisation the picture changes. Surveying 400 of a 500-person workforce is not wasteful; the finite population correction cuts the required sample sharply when you are sampling a large share of the group, and a census of everyone removes sampling error entirely. The other half of the job is non-response.
An internal survey that historically gets a 30% response rate needs roughly three invitations for every completed response, and the people who do not reply are rarely a random subset. Sample size controls sampling error only, so a large sample with a 15% response rate can still be badly biased, which is why researchers chase response rates as hard as they chase numbers.
Worked examples
95% confidence, ±5% margin, population of 10,000
370 completed responses needed, against 385 if the population were unlimited.
99% confidence and a tighter ±3% margin in a population of 100,000
1,810 completed responses, showing how fast a tighter margin adds cost.
A 500-member club at 95% confidence and ±5%, with a 40% response rate
218 completed responses, which means sending 545 invitations.
When people use it
- Sizing a customer or employee survey before fieldwork starts
- Checking whether an existing sample supports the claim a report makes
- Deciding how many invitations to send given a known response rate
- Planning acceptance sampling where each test costs money or destroys a unit
- Showing a stakeholder why a tighter margin of error quadruples the budget
Tips
- Use 50% for the expected split when you have no prior information. It maximises the variance term and gives the safest sample size.
- Halving the margin of error quadruples the sample, so ±2.5% costs four times what ±5% costs.
- The finite population correction only bites when your sample is a large share of the population, roughly above one in twenty.
- Sample size is calculated on completed responses. Divide by your expected response rate to get the number of people to invite.
- If you plan to report on subgroups, each subgroup needs its own sample size; a 400-person survey cut four ways gives 100 per cell.
- Clustered or stratified designs change the effective sample size. Multiply by the design effect before trusting a simple random figure.
What it assumes, and where it stops being accurate
- Simple random sampling from the population. Cluster, quota and convenience samples need a design effect applied on top.
- The variable of interest is a proportion. Estimating a mean requires the standard deviation form of the formula instead.
- Non-response is assumed to be random, which is rarely true. A low response rate can bias results no matter how large the sample.
- The normal approximation to the binomial is used, which becomes unreliable when the expected split is very close to 0% or 100%.
- Measurement error, question wording effects and coverage gaps in the sampling frame are not captured by the margin of error at all.
Terms used on this page
- Margin of error
- The half-width of the confidence interval around a survey result, in percentage points.
- Confidence level
- The proportion of repeated samples whose interval would contain the true value.
- Finite population correction
- An adjustment that reduces the sample when it is a large share of a small population.
- Design effect
- A multiplier accounting for the loss of precision in clustered or stratified sampling.
Questions people ask
- What confidence level should I use?
- Ninety-five per cent is the working standard across published research and commercial surveys, and reviewers expect it unless you say otherwise. Use 99% when the consequences of being wrong are serious, such as clinical or safety decisions, and accept the larger sample it demands. Ninety per cent is defensible for exploratory work where the result guides rather than decides.
- Why does the population size barely change the answer?
- Because precision depends on the number of responses, not on the fraction of the population they represent. The finite population correction only reduces the sample meaningfully when you are surveying a large share of a small group. Above roughly 100,000 people the required sample is effectively the same whether the population is a city or a continent.
- What does a margin of error of ±5% actually mean?
- That if you repeated the survey many times with fresh samples, about 95 of every 100 intervals built as the result plus or minus five points would contain the true population value. A single result of 52% therefore supports a true figure anywhere between 47% and 57%, which is why small leads within the margin are not real findings.
- Why use 50% for the expected split?
- Because p(1 − p) reaches its maximum at 0.5, so that assumption produces the largest and therefore safest sample size. If you have reliable prior data showing the split is nearer 20/80, entering it will reduce the sample required by about a third. Guessing an extreme split without evidence risks an undersized survey.
- How do I allow for people who do not respond?
- Divide the required sample by your expected response rate, which this calculator does for you in the invitation figure. A 30% response rate means inviting more than three people for every completed response. Bear in mind that this only fixes the count; it does not fix the bias caused by non-responders differing from responders.
- Does this work for estimating an average rather than a percentage?
- Not directly. For a mean you need n = (z × σ / E)², where σ is the population standard deviation and E is the margin of error in the same units as the data. The structure is the same, but the variance term comes from the standard deviation instead of from p(1 − p).
Use this with
Sample Size Calculator answers one part of the question. These pick up where it stops: Z-Score Calculator, Standard Deviation Calculator, Probability Calculator and Percentage Calculator.
Where these figures come from
This calculation is jurisdiction-neutral arithmetic. No country-specific rules are applied.
- American Association for Public Opinion Research. Standards for survey sampling and margin-of-error reporting
- Pew Research Center: methods. Practical survey methodology, weighting and non-response
- NIST/SEMATECH e-Handbook of Statistical Methods. Reference treatment of descriptive statistics and hypothesis testing
Sample Size Calculator produces an estimate for general information. It is not financial, tax, medical, legal or engineering advice. read the full disclaimer. Spotted a wrong number? Report it and it gets fixed. The form picks up this calculator and the inputs you used.