Survey sample size calculator

Sample size is the number of completed responses you collect and analyse, written as n. It is the number that sets your margin of error, which is how far a survey result can sit from the figure you would get by asking the whole population. It counts finished responses, not invitations sent and not the size of your list, and a larger n buys precision, not accuracy.

Sample size calculator

Nothing is hidden behind a signup and nothing is sent anywhere. The calculator runs in your browser, starts with a worked answer already on screen, and prints the formula and the arithmetic it used so you can check it or reproduce it by hand.

What sample size means

Sample size is the number of people who complete your survey. It is written as n in every formula on this page, it is the one quantity you buy with fieldwork rather than choose on paper, and it is the figure your margin of error is calculated from.

Three counts get mistaken for it, and telling them apart is most of what a sample plan does:

• Population (N): everyone you want to describe. A customer list, a headcount, a country.
• Invitations: how many people you contact. Always larger than n, because not everyone answers.
• Sample size (n): how many completed responses you hold when fieldwork ends. This is the number the formula sizes, and the only one of the three that changes your margin of error.

Sample size buys precision, which is the width of the plus or minus band around each result. It does not buy accuracy, which is whether the people who answered resemble the people who did not. A large convenience sample can sit further from the truth than a small random one, and no value of n repairs that. Where sample size planning goes wrong covers the difference, and how you draw the sample is what decides it.

The short answer, if you only want a number

The sample size formula, and how to do it by hand

For a survey that reports a percentage, the standard starting point is the sample size for a population large enough to treat as unlimited:

n0 = (z^2 x p x (1 - p)) / E^2

Reading it left to right:

• n0 is the required completed responses, before any population correction.
• z is the z score for your confidence level. Use 1.96 for 95 percent.
• p is the proportion you expect to measure, as a decimal. Use 0.5 if you do not know.
• E is the margin of error as a decimal. Five percentage points is 0.05.

The symbol for sample size is n. Some sources write the uncorrected figure as n0 and the corrected one as n, which is the convention used here.

Worked example: from 385 to an invitation list

Scenario. You want the percentage of customers who would recommend you. You want 95 percent confidence and plus or minus 5 points. You have no prior estimate, so p = 0.5.

1. Pick your confidence level and margin of error

   95 percent confidence sets z = 1.96. Plus or minus 5 percentage points is E = 0.05.

2. Choose an expected result

   p = 0.5, because there is no prior estimate and 0.5 produces the largest sample. Nothing you learn later can leave you short.

3. Apply the sample size formula

   n0 = (1.96^2 x 0.5 x 0.5) / 0.05^2

   n0 = (3.8416 x 0.25) / 0.0025 = 0.9604 / 0.0025 = 384.16

   Round up: 385 completed responses. Always round up. You cannot collect a fraction of a survey.

4. Correct for a small population

   If those customers are a list of 1,000 rather than an open market, apply the correction: n = 384.16 / (1 + 383.16 / 1,000) = 384.16 / 1.38316 = 277.7, so 278 completes. The correction only bites once the sample is a large share of the population.

5. Turn completes into invitations

   At a 30 percent response rate, invitations = 278 / 0.30 = 926.7, so invite 927 people. Completes are what the formula sizes. Invitations are what you actually send, and they can never exceed the population you are drawing from.

At 95 percent confidence with p = 0.5, the whole formula collapses to 0.9604 divided by E squared. That single shortcut answers most sample size questions in one line, and it is why 385 keeps appearing in sample size tables you find elsewhere.

Sample size table for common population sizes

These figures are computed with the formula above at 95 percent confidence, plus or minus 5 points and p = 0.5, then corrected for the population. The last two columns are our own check against the table published by Krejcie and Morgan in 1970, which is the table most methods chapters cite.

How we checked it. We ran the formula at 21 population sizes and compared each result with the published Krejcie and Morgan value. Twelve matched exactly. The other nine differed by exactly one response, which comes from where each method rounds, not from a disagreement about the arithmetic. If a supervisor asks why your number differs from the table by one, that is the reason.

Worth knowing if you are citing that table: Krejcie and Morgan did not derive their own formula. Their 1970 paper is about 400 words plus one table, and its single reference credits the formula to a 1960 research bulletin from the National Education Association. Their own summary of the pattern is still the best one line description of it, that as the population grows the required sample increases at a diminishing rate and settles at slightly more than 380 cases.

Choosing your inputs, one at a time

Four numbers drive the answer. Three of them are decisions, not facts, and each one has a defensible way to pick it.

Confidence level: how often the method is right

Confidence level describes the method, not this particular survey. At 95 percent confidence, intervals built this way contain the true value 95 times out of 100 in the long run. It is not a 95 percent chance that your specific answer is correct.

Use 95 percent unless you have a reason not to. Moving to 99 percent raises the sample from 385 to 664 for the same precision, which is a 72 percent cost increase to buy very little. The related idea in hypothesis testing is alpha, which is why people mix the two up. If that distinction is fuzzy, read statistical significance and what p-values mean before you pick a number.

Margin of error: the decision variable

This is the one to think hardest about, because it is the only input that ties the sample to a decision. Pick it by asking what gap would change what you do.

If you act differently when satisfaction is 72 percent rather than 74 percent, you need a tight margin. If you only act at 72 versus 82, plus or minus 5 points is plenty. Broad top-line reporting usually lands at 5 points for a pulse check or 3 points for tracking. Below 2 points the cost climbs fast.

Population size: mostly irrelevant, until it is not

For big populations the required sample barely moves. Ten thousand people need 370 responses. A hundred thousand need 383. Population size only bites when your sample is a meaningful share of it, which is the case for employee lists, member registers and school cohorts. That is what the next section handles.

Expected result (p): use 0.5 when you do not know

For a percentage question, the uncertainty is driven by p times (1 minus p). That term is largest at p = 0.5, so 0.5 is the conservative choice and produces the largest sample. It is the right default when you have no prior estimate.

If you do have one, use it, because the saving is large. At 95 percent confidence and plus or minus 5 points, p = 0.5 asks for 385 responses and p = 0.9 asks for 139. Past surveys, a pilot, or a published benchmark are all fair sources for that estimate. Noisy questions inflate the spread too, so clearer wording can stabilise a result without any extra responses. Writing better survey questions is often cheaper than collecting more of them.

Small populations: the finite population correction

The base formula assumes you are drawing from a pool deep enough that removing people from it changes nothing. Once your sample is a real slice of the population, that assumption overcharges you. The finite population correction gives the responses back.

n = n0 / (1 + ((n0 - 1) / N))

• n is the corrected number of completes you need.
• n0 is the uncorrected figure from the formula above.
• N is the population size.

The usual trigger is that n0 is more than about 5 percent of N. Below that the correction changes almost nothing, so it is not worth explaining to anyone.

Worked example: an employee survey

Scenario. You have 1,200 employees. You want 95 percent confidence, plus or minus 5 points, and you use p = 0.5.

• Uncorrected, n0 = 384.16.
• Corrected: n = 384.16 / (1 + (383.16 / 1,200)) = 384.16 / 1.3193 = 291.2.
• Round up: 292 completed responses.

Without the correction you would chase 385 responses from a list of 1,200, which is 32 percent of everyone. With it, 292 buys the same precision. On a list that size that is a month of chasing you do not have to do.

Margin of error: the same formula, run backwards

Margin of error is the plus or minus band around a survey result that comes from having asked some people instead of all of them. If 62 percent of 385 respondents pick an option and the margin of error is 5 points, you report 62 percent and say the plausible range runs from 57 to 67 percent.

Rearranging the sample size formula to solve for E gives:

E = z x square root of (p x (1 - p) / n)

With a finite population, multiply that by the square root of (N - n) divided by (N - 1). Both versions are in the calculator below, which switches automatically when you tick or untick the population box.

Does margin of error increase with sample size?

It shrinks, and it shrinks with the square root of n. That square root is the whole economics of surveys. Four hundred responses give you half the error of one hundred. To halve it again you need 1,600. Each further decimal place of precision costs roughly four times the fieldwork, which is why 385 is such a common landing point: it is where cost and precision stop arguing.

Four terms people mix up

Standard error, margin of error, confidence level and confidence interval get used as though they were interchangeable. They are four different things, and the difference matters when someone questions your number.

The short version: standard error is a property of your data, confidence level is a choice you made, margin of error is the two multiplied together, and the confidence interval is what you print in the report.

What the margin of error does not cover

It covers random sampling error and nothing else. It does not cover people who ignored the invitation, a list that was missing part of the population, or a question that half your respondents read differently from the way you meant it. A survey can have a tidy margin of error and still be wrong. See response bias and nonresponse bias for the errors that no sample size fixes.

The professional body for survey research puts this more bluntly than most vendors will. AAPOR states that there is no such thing as a measurable overall margin of error for a poll, because surveys carry other errors that cannot be measured at all, from how the questions were written to how the interviews were run. What you can compute is the sampling piece. That is the piece the calculator gives you, and it is worth being clear that it is a piece.

Two usage rules from the same source, both of which most survey writing gets wrong:

• It is percentage points, not a percentage. Write "plus or minus 3 percentage points", not "plus or minus 3 percent". They mean different things.
• It does not apply to opt-in samples at all. A margin of error assumes every person had a known, non-zero chance of selection. An open link, a panel of volunteers or a social media post does not meet that. Publishing a margin of error next to results from an open link is claiming a precision you did not buy.

Turning completes into invitations

The formula gives you finished responses. Fieldwork gives you a response rate. The gap between them is where sample plans usually fail, and the arithmetic is one line:

Invitations = required completes / expected response rate

Use the rate as a decimal, so 30 percent is 0.30, and round up. For 292 completes at 35 percent, that is 292 divided by 0.35 = 834.3, so invite at least 835 people.

Picking a response rate you can defend

Your own history beats any published benchmark, as long as it is the same audience, the same channel and a similar length of survey. If you have nothing to go on, plan three scenarios at 20, 30 and 40 percent and treat the invitation list as an adjustable lever during fieldwork. Reminders, a longer window and a wider list are all things you can change midway. The completes target is not.

Most benchmark tables on the internet quote one average with no source. There is no single average. Here is what the primary sources actually report, so you can cite the row that matches your situation instead of a number somebody made up.

Two things fall out of that table. First, the size of the group you are surveying predicts the rate better than almost anything else you control. A 300 person company and a 300,000 person agency are not running the same survey. Second, the widely repeated claim that response rates are collapsing everywhere is only true in some places. Telephone polling did collapse. Published organisational research went the other way, from 48 percent to 68 percent over fifteen years.

One more piece of planning arithmetic worth borrowing: the standard extension guidance is to add about 10 percent to your invitation list for people you simply cannot reach, and about 30 percent for people who could answer and do not.

What our own template corpus says about length

A response rate borrowed from someone else's survey travels badly, and length is one of the few things on that list you fully control. Here is a figure we can produce and a competitor cannot, because it comes from counting our own published corpus rather than asking practitioners what they think they do.

All 87 survey templates published on this site are 14 questions or shorter, and 86 of the 87 are 10 questions or shorter. Together they hold 799 questions. The median template is 9 questions, the shortest is 7, and exactly one runs past 10.

By answer format, 355 of the 799 questions are rating scales, 281 are single choice lists, and 163 ask the reader to type something. That last group is the one that costs you completions, so we read it by hand instead of trusting the stored field. Of the 163, 137 are multi line comment boxes. The other 26 are single line boxes, and reading them showed that 22 are short identifier or classification fields: a name, an email address, a date, a department, a job title. Only 4 are substantive short answers. So the honest count of genuinely open ended questions is 141, or roughly one question in six, not the one in five the raw field would have handed us.

How we counted, and how we checked it. We parsed the stored question list of every survey template live on this site on 21 August 2026, kept the entries carrying question text, then fetched each published page and compared the two sets rather than trusting our own database on its own. All 87 pages returned HTTP 200, while a slug we know is dead returned 410 in the same run, so the check was discriminating rather than uniformly green. 785 of the 799 questions, or 98 percent, appear on the published page. Six of those matches differ from the database only in which apostrophe character is used, and we counted them as matches, which we would rather disclose than quietly round away. Eighty six of the 87 templates verified in full. The one that did not is our longest, at 14 questions, and its published page currently renders no preview questions at all, so its questions could not be confirmed from the outside.

What this does not tell you is the response rate those templates achieve. We went looking. The survey response records we hold stop in 2015 and belong to an earlier product, so any platform response rate we published from them would describe a different decade and a different tool. We are not going to put that number on this page, and you should treat any vendor who quotes one without showing the method the same way. The length figures above are ours and you can check them against the template pages. The response rates in the table above are other people's, each with a source you can open.

The practical use is narrow and real. If you are borrowing a response rate for the invitation count in the calculator, borrow it from a survey about the length of the one you are about to send, to a similar audience on a similar channel. Our own published instruments sit between 7 and 10 questions, which is shorter than most drafts start out.

More invitations do not buy you less bias

A low response rate creates two problems that look like one. The first is that you finish short of your target, so your margin of error is wider than you planned. The second is that the people who answered may differ from the people who did not, in ways that matter to your result.

Inviting more people fixes the first problem. It does not reliably fix the second. Groves and Peytcheva pooled 59 studies that measured actual nonresponse bias against known truth, and found only a small relationship between a survey's response rate and how biased it turned out to be. Pew tested the same idea directly: they ran one questionnaire twice, once with standard methods that got a 9 percent response rate and once with a high effort protocol that got 22 percent. Of the 40 questions the two surveys shared, 28 came back within 2 points of each other.

The honest counter-case. That finding does not generalise to every topic, and it is worth knowing where it breaks. In satisfaction surveys the relationship is real and large. A study of 717 hospital patient satisfaction surveys across 164 Swiss hospitals found mean satisfaction score and response rate correlated at r = 0.61, because delighted and furious people answer and the ambivalent middle does not. If you run CSAT or NPS, a rising response rate can move your score on its own, and a low one probably flatters you.

So: chase response rate when you are measuring how people feel about you. Do not treat it as a certificate of validity when you are measuring anything else.

Sample size for quantitative research: what a methods chapter needs

There is no single good sample size for quantitative research, and any source that gives you one without asking what you are measuring is guessing. What gets a methods chapter approved is not a large number. It is a stated number, with the confidence level, margin of error, assumed proportion and population written next to it, and a named formula attached.

Supervisors tend to expect one of four named approaches. They mostly agree with each other, and it is worth knowing which one you are being asked for.

Cochran, and why everything else is a shortcut of it

Cochran's formula is the one at the top of this page: n0 = z squared times p times (1 minus p), divided by E squared, followed by the finite population correction when the population is small. The other named methods fix some of those inputs to make the arithmetic shorter. Once you know that, the family stops looking like four competing rules and starts looking like one rule with three presets.

Yamane and Slovin: use with your eyes open

The formula n = N / (1 + N x e^2) appears in thousands of theses, usually credited to Slovin. It is quick: for N = 1,000 and e = 0.05 it gives 286.

Three things are worth knowing before you cite it.

• It is Cochran with the settings welded shut. Working backwards, the formula is the proportion formula with p fixed at 0.5, confidence fixed at 95 percent, and z rounded from 1.96 to 2. It cannot answer a question at any other confidence level, and it has no input at all for how variable your population is. Two populations of the same size but wildly different spread get the same answer, which cannot be right.
• That rounding is why its numbers run high. With z = 2 the formula's ceiling for an unlimited population is exactly 1 divided by e squared, which is 400. Cochran's ceiling is 384.16, because 1.96 squared is 3.8416 and not 4. Every Yamane figure in the table above is a few percent larger than the Cochran figure for exactly that reason. If you have ever wondered why two textbooks give you two different sample sizes for the same study, that is usually the answer.
• The attribution does not hold up. A statistics journal traced the formula's history and concluded that no person named Slovin can be found behind it, and that the oldest traceable source is Taro Yamane's 1967 textbook. Wolfram MathWorld describes it as an ad hoc formula lacking mathematical rigour. If your department expects it, use it and cite Yamane rather than Slovin. If you have a choice, use Cochran, because it survives the question of where the 95 percent came from.

Rules of thumb, honestly graded

These circulate widely. Some are useful, some are folklore, and it helps to know which is which.

• At least 30 per group. Reasonable as a floor for group comparisons, not as a target. It comes from where sampling distributions start behaving, not from any precision requirement.
• Ten cases per variable in a regression. A working heuristic, widely used and widely criticised. State it as a heuristic, never as a rule.
• Ten percent of the population. This one is simply wrong and the table above shows why. Ten percent of 100,000 is 10,000 responses, when 383 gives you the same precision.
• Always 385. Right only for a large population at 95 percent and 5 points. Quote it with those conditions attached or it means nothing.
• Small n for usability and product studies. Nielsen Norman Group publishes the honest version of this: 40 participants gets you a yes or no measure to plus or minus 15 points at 95 percent confidence, and 47 gets you an average to the same precision. Those numbers are small because the precision claimed is loose. Say the precision out loud and small samples stop being embarrassing.

If your study compares groups rather than estimating one percentage, sample size becomes a power question and this is not the right calculator. You need an effect size you are willing to defend, plus alpha and power. G*Power and OpenEpi are the free tools most supervisors expect, and both are free. Saying where a tool stops is more useful than pretending it does everything.

Qualitative work is sized differently

Interviews and focus groups are not sized with a margin of error, because they are not estimating a population percentage. The usual argument is saturation: you keep going until new participants stop producing new themes, then report where that happened and how you judged it. Running a qualitative study through this calculator produces a number that looks precise and means nothing.

If you need results by subgroup, size for the subgroup

Most surveys are asked to report by region, role, tenure or product line. This is where sample plans quietly break, and the rule is short.

Your margin of error applies to a group only if that group has that many completes. If you want plus or minus 5 points for each of six regions, you need roughly 385 completes in each region, not 385 in total. That is a survey six times the size you thought you were planning.

Two honest ways out:

• Oversample the small groups. Deliberately invite a higher share of the smaller segments so each one reaches its own target. You weight the results back afterwards.
• Accept wider bands for the breakouts. Keep one strong overall number and publish each subgroup with its own, larger margin of error next to it. This is usually the right answer.

The calculator has a field for this. Set Subgroups to report separately to the number of groups you owe a result for, and it sizes each group properly and totals them.

Scope it before you field. List the segments you must report, not the ones that would be nice to have. Estimate the completes each one can realistically produce, using demographic questions to identify them. If a segment can only yield 80 responses, do not promise plus or minus 5 points for it. At 80 responses the band is plus or minus 11 points.

AAPOR uses a worked example that is worth memorising. A survey of 1,000 adults carries an overall margin of error of about plus or minus 3 points. If 200 of those people are the subgroup you actually want to talk about, that subgroup's margin of error is plus or minus 6.9 points, more than double. The headline figure on the front of the report is not the figure that governs the chart on page nine.

Published guidance on minimum cell sizes is older and looser than the formulas, and it is fair to treat it as a floor rather than a target: roughly 100 responses for a subgroup you will report on properly, and 20 to 50 for a minor one you will mention in passing.

Where sample size planning goes wrong

Sample size manages sampling error, the random noise from not asking everyone. Real surveys also carry systematic error, which no sample size fixes and which a large sample can disguise.

Treating a convenience sample like a probability sample

Every formula here assumes each person had a known, non-zero chance of being selected, and that answering was not strongly tied to what you are measuring. An open link on social media meets neither condition. You can still compute a margin of error from the responses, and it will look reassuring, and it will not mean what it says. Be explicit about which kind of sample you actually drew.

Reading a bigger n as better measurement

A large sample makes a noisy measurement precisely noisy. If a question is ambiguous, double-barrelled or leading, more responses will not rescue it. High skip rates and inconsistent answers are signs to fix the instrument first. Rewriting one bad question often does more than tripling the sample.

Choosing precision without a decision behind it

Precision only has a right answer once you know what it is for. A sample size is easy to defend when you can finish this sentence: we need to tell apart outcomes that differ by X points. It is impossible to defend when the only rationale is that a bigger number felt safer. Tie it back to your research design and to what you will do with the result.

The numbers on this page, and how good they are

Every figure quoted above is either arithmetic you can reproduce in the calculator or a citation you can check. Grade A means replicated or directly computable, B means a single strong study, C means widely repeated but contested.

The justification template

Most sample size arguments fail because the assumptions were never written down. The Copy the justification button in the calculator writes this out with your own numbers already in it. If you would rather assemble it yourself, these are the nine things a reviewer will look for.

• warning
  Population (N): define the population and state the count. For example, all active customers in the last 90 days, N = 48,200.
• warning
  What you are measuring: a proportion (a percentage) or a mean (an average rating). The formula here sizes proportions, which covers most reported KPIs.
• warning
  Confidence level: the level you chose and why. Convention and risk tolerance are both acceptable reasons if you say so.
• warning
  Margin of error (E): your target precision overall, and separately for any subgroup you promised to report.
• warning
  Assumed proportion (p): the value used, and where it came from. State plainly if it is 0.5 because nothing better was available.
• warning
  Finite population correction: whether you applied it and what triggered it, such as n0 exceeding 10 percent of N.
• warning
  Completes target (n): the final required number of completed responses after every adjustment.
• warning
  Expected response rate: the rate assumed, the history it came from, and the resulting invitation count.
• warning
  How you will select people: simple random, stratified, census, or convenience, plus any oversampling for key segments. Name it honestly.

Once the survey is in the field, keep watching completes by segment rather than the total. A healthy overall number can hide a segment that will never reach its target, and that is a problem you can still fix at week one and cannot fix at week four. When you report, pair every percentage with its margin of error, and read how to analyse survey data before treating a two point move as real.

Frequently Asked Questions

References