To get a 5% margin of error at 95% confidence you need 385 completed responses when the population is large, and fewer when it is small: 370 for a population of 10,000, 278 for 1,000, 218 for 500 and 80 for 100. Those counts come from Cochran's formula with the finite population correction, a simple random sample, and the most cautious assumption that the answer you are measuring splits 50/50.
Statistics explainer, not survey design advice for a specific study. Every number in the tables below was computed by script from the formulas shown, with z = 1.645, 1.96 and 2.576 for 90%, 95% and 99% confidence, p = 0.5 unless a table says otherwise, and sample sizes rounded up to the next whole respondent. Formulas are from the NIST/SEMATECH e-Handbook of Statistical Methods and Penn State's STAT 506 course notes, both checked October 30, 2026.
People searching this question usually have a fixed population in front of them: 600 employees, 240 members of an association, 1,800 customers on a mailing list, a school of 450 students. The famous 385 is the answer for an unlimited population, and applying it to a list of 240 people is how surveys end up asking for more responses than they need, or concluding that a sample is hopeless when it is fine. The tables here are laid out by population size first, because that is the number you already know.
Two formulas produce every number on this page
The starting point is the sample size for a proportion when the population is so large that sampling a few hundred people barely dents it. Penn State's STAT 506 notes give it, for when "the finite population correction can be ignored," as:
Here z is the normal critical value for your confidence level, p is the proportion you expect to see (0.5 if you have no idea), and d is the margin of error written as a decimal, so 5% is 0.05. Many textbooks call this Cochran's formula after William Cochran's sampling text, and they write e where Penn State writes d. The symbols differ; the arithmetic does not.
The critical values come from the standard normal table. The NIST e-Handbook, section 1.3.6.7.1, lists the commonly used critical values as ±1.645, ±1.960 and ±2.576, which leave 5%, 2.5% and 0.5% in each tail. Two tails of 5% each make a 90% interval, two of 2.5% make 95%, and two of 0.5% make 99%.
For a finite population of size N, the same Penn State lesson derives the corrected size from the variance of a sample proportion drawn without replacement:
Divide the top and bottom by p(1 - p) and it becomes the version most calculators print:
Both lines give identical answers; the second just reuses the unlimited-population number you already computed. Our sample size calculator uses exactly this second form, with the same three z values, so its output matches every cell below.
Why does the population size matter at all? Because once you have sampled a large share of a small group, there is not much left that could surprise you. Penn State writes the variance of the sample proportion as the infinite-population variance multiplied by (N - n) / (N - 1). When you survey 400 people out of 1,000, that factor is 600/999, and its square root, 0.775, shrinks the margin of error by about 22% compared with drawing 400 from a population of millions.
Sample size at 95% confidence, by population and margin of error
This is the table the target question asks for. Rows are population sizes, columns are the margin of error you want, and each cell is the number of completed responses needed at 95% confidence with p = 0.5.
| Population | ±10% | ±7% | ±5% | ±4% | ±3% | ±2% | ±1% |
|---|---|---|---|---|---|---|---|
| 50 | 34 | 40 | 45 | 47 | 48 | 49 | 50 |
| 100 | 50 | 67 | 80 | 86 | 92 | 97 | 99 |
| 200 | 66 | 100 | 132 | 151 | 169 | 185 | 196 |
| 300 | 73 | 119 | 169 | 201 | 235 | 267 | 291 |
| 500 | 81 | 142 | 218 | 274 | 341 | 414 | 476 |
| 1,000 | 88 | 165 | 278 | 376 | 517 | 707 | 906 |
| 2,000 | 92 | 179 | 323 | 462 | 697 | 1,092 | 1,656 |
| 5,000 | 95 | 189 | 357 | 537 | 880 | 1,623 | 3,289 |
| 10,000 | 96 | 193 | 370 | 567 | 965 | 1,937 | 4,900 |
| 100,000 | 96 | 196 | 383 | 597 | 1,056 | 2,345 | 8,763 |
| 1,000,000 | 97 | 196 | 385 | 600 | 1,066 | 2,396 | 9,513 |
| Unlimited | 97 | 196 | 385 | 601 | 1,068 | 2,401 | 9,604 |
The unlimited-population values before rounding are 96.04, 196.00, 384.16, 600.25, 1,067.11, 2,401.00 and 9,604.00. The ±3% figure of 1,068 is the same "round up to 1068" that Penn State reaches in its presidential approval example with the conservative p = 0.5.
What the 95% table says once you stare at it
Small groups need most of their members. At ±5%, a population of 50 needs 45 responses, which is 90% of everyone. A population of 100 needs 80. In groups this small the question is less "how many should I sample" than "can I reach nearly everybody," and if you can't, a looser margin like ±10% (34 of 50, or 50 of 100) is the honest target.
Large populations stop mattering. Between a population of 100,000 and one of unlimited size, the ±5% column moves from 383 to 385. That flat line is why a national poll and a poll of one mid-sized city need roughly the same sample for the same margin. The population only pulls the number down when the sample would be a noticeable fraction of it; as a rough reading of the table, that starts when the unlimited-population answer is more than about 5% of N.
Halving the margin roughly quadruples the sample. Because d is squared in the denominator, going from ±5% to ±2.5% takes the unlimited number from 385 to 1,537. Going from ±2% to ±1% takes it from 2,401 to 9,604. Each step of precision costs far more than the one before, which is why most opinion surveys settle around ±3% to ±5%.
A million is not quite unlimited at ±1%. The ±1% cell for a population of 1,000,000 is 9,513, while the unlimited row says 9,604. At that tight margin the sample is about 1% of the population, enough for the correction to save 91 responses. If you are planning a survey of a very large but definitely finite group at ±1%, use the actual population, not the unlimited row.
The same grid at 90% and 99% confidence
Changing the confidence level changes only z, and z is squared. Moving from 95% to 99% multiplies the unlimited-population sample by (2.576 / 1.96)², which is 1.727. Dropping to 90% multiplies it by (1.645 / 1.96)², which is 0.704. In small populations the finite correction softens both effects.
90% confidence (z = 1.645, p = 0.5):
| Population | ±10% | ±7% | ±5% | ±4% | ±3% | ±2% | ±1% |
|---|---|---|---|---|---|---|---|
| 50 | 29 | 37 | 43 | 45 | 47 | 49 | 50 |
| 100 | 41 | 59 | 74 | 82 | 89 | 95 | 99 |
| 200 | 51 | 82 | 116 | 136 | 159 | 179 | 195 |
| 300 | 56 | 95 | 143 | 176 | 215 | 255 | 288 |
| 500 | 60 | 109 | 176 | 230 | 301 | 387 | 466 |
| 1,000 | 64 | 122 | 214 | 298 | 430 | 629 | 872 |
| 2,000 | 66 | 130 | 239 | 350 | 547 | 917 | 1,544 |
| 5,000 | 67 | 135 | 257 | 390 | 654 | 1,264 | 2,876 |
| 10,000 | 68 | 137 | 264 | 406 | 700 | 1,447 | 4,036 |
| 100,000 | 68 | 138 | 270 | 422 | 747 | 1,664 | 6,337 |
| 1,000,000 | 68 | 139 | 271 | 423 | 752 | 1,689 | 6,720 |
| Unlimited | 68 | 139 | 271 | 423 | 752 | 1,692 | 6,766 |
99% confidence (z = 2.576, p = 0.5):
| Population | ±10% | ±7% | ±5% | ±4% | ±3% | ±2% | ±1% |
|---|---|---|---|---|---|---|---|
| 50 | 39 | 44 | 47 | 48 | 49 | 50 | 50 |
| 100 | 63 | 78 | 88 | 92 | 95 | 98 | 100 |
| 200 | 91 | 126 | 154 | 168 | 181 | 191 | 198 |
| 300 | 108 | 160 | 207 | 233 | 259 | 280 | 295 |
| 500 | 125 | 203 | 286 | 338 | 394 | 447 | 486 |
| 1,000 | 143 | 254 | 400 | 510 | 649 | 806 | 944 |
| 2,000 | 154 | 290 | 499 | 684 | 960 | 1,350 | 1,785 |
| 5,000 | 161 | 318 | 586 | 859 | 1,347 | 2,268 | 3,843 |
| 10,000 | 164 | 328 | 623 | 940 | 1,557 | 2,932 | 6,240 |
| 100,000 | 166 | 338 | 660 | 1,027 | 1,810 | 3,983 | 14,230 |
| 1,000,000 | 166 | 339 | 664 | 1,036 | 1,840 | 4,131 | 16,319 |
| Unlimited | 166 | 339 | 664 | 1,037 | 1,844 | 4,148 | 16,590 |
At 99% and ±1% the formula asks for all 100 members of a population of 100. That is a census, and it is the formula telling you that the precision you asked for is not available from anything less than everyone.
A practical reading of the three tables together: for a population of 1,000, the choice between 90%, 95% and 99% confidence at ±5% is a choice between 214, 278 and 400 responses. If reaching each extra person is cheap, the higher level costs little. If each response takes a phone call or a site visit, the 186-response gap between 90% and 99% is real money and time, and 95% is the usual compromise for a reason.
Working a population of 400 by hand
Suppose a residents' association has 400 households and wants to report the share who support a proposal within ±5%, at 95% confidence, with no prior idea of the answer. This is a hypothetical example with stated inputs: N = 400, d = 0.05, z = 1.96, p = 0.5.
- Unlimited-population size. 1.96² is 3.8416. Multiply by 0.5 × 0.5 = 0.25 to get 0.9604. Divide by 0.05² = 0.0025. The result is n₀ = 384.16. Keep the decimals; do not round yet.
- Population adjustment. Subtract 1 from n₀ and divide by the population: 383.16 / 400 = 0.9579.
- Divide. 1 + 0.9579 = 1.9579, and 384.16 / 1.9579 = 196.21.
- Round up. You need 197 completed responses.
Penn State's own worked examples round the same way ("We will always round this up"), because rounding down gives a sample that falls just short of the margin you promised.
To confirm it, run the reverse formula from the next section on the answer. With 197 responses out of 400, the margin of error is 1.96 × √(0.25 / 197 × 203 / 399) = ±4.98%, just inside the ±5% target. Had you rounded down to 196, you would sit slightly above it.
The same steps for 1,200 households give (384.16 - 1) / 1,200 = 0.3193, then 384.16 / 1.3193 = 291.18, so 292 responses. Three times the population needs only about 1.5 times the sample. That ratio is the finite population correction at work, and it is the reason the table bends so sharply below a population of about 2,000.
A common slip is to round n₀ to 385 before applying the correction. For a population of 400 it makes no difference here (the result is still 197), but in a few cells of the tables above it shifts the answer by one respondent. The tables use the unrounded n₀ throughout, as our calculator does.
Going backwards: the margin of error a sample already gives you
Often the sample came first. A survey went out, 212 people answered, and someone wants to know how precise the result is. Solving the Penn State variance expression for the margin gives:
Leave off the second fraction when the population is very large. With p = 0.5 this is the "maximum margin of error" that polling firms quote, because no observed percentage can produce a wider interval than 50% does.
| Responses (n) | 90%, large pop. | 95%, large pop. | 99%, large pop. | 95%, N = 500 | 95%, N = 1,000 | 95%, N = 5,000 |
|---|---|---|---|---|---|---|
| 30 | ±15.0% | ±17.9% | ±23.5% | ±17.4% | ±17.6% | ±17.8% |
| 50 | ±11.6% | ±13.9% | ±18.2% | ±13.2% | ±13.5% | ±13.8% |
| 100 | ±8.2% | ±9.8% | ±12.9% | ±8.8% | ±9.3% | ±9.7% |
| 150 | ±6.7% | ±8.0% | ±10.5% | ±6.7% | ±7.4% | ±7.9% |
| 200 | ±5.8% | ±6.9% | ±9.1% | ±5.4% | ±6.2% | ±6.8% |
| 250 | ±5.2% | ±6.2% | ±8.1% | ±4.4% | ±5.4% | ±6.0% |
| 300 | ±4.7% | ±5.7% | ±7.4% | ±3.6% | ±4.7% | ±5.5% |
| 400 | ±4.1% | ±4.9% | ±6.4% | ±2.2% | ±3.8% | ±4.7% |
| 500 | ±3.7% | ±4.4% | ±5.8% | census | ±3.1% | ±4.2% |
| 600 | ±3.4% | ±4.0% | ±5.3% | n/a | ±2.5% | ±3.8% |
| 800 | ±2.9% | ±3.5% | ±4.6% | n/a | ±1.6% | ±3.2% |
| 1,000 | ±2.6% | ±3.1% | ±4.1% | n/a | census | ±2.8% |
| 1,500 | ±2.1% | ±2.5% | ±3.3% | n/a | n/a | ±2.1% |
| 2,000 | ±1.8% | ±2.2% | ±2.9% | n/a | n/a | ±1.7% |
| 2,500 | ±1.6% | ±2.0% | ±2.6% | n/a | n/a | ±1.4% |
| 5,000 | ±1.2% | ±1.4% | ±1.8% | n/a | n/a | census |
Inputs: p = 0.5, the z values above, and margins rounded to one decimal. "Census" means the sample is the whole population, so there is no sampling error left to report.
Some numbers worth carrying around from this table: 100 responses give about ±10% at 95% confidence, 400 give about ±5%, and 1,000 give about ±3%. The population columns show how much a small population helps. Four hundred responses from a population of 500 carry a margin of only ±2.2%, against ±4.9% from a large population.
When this margin is too optimistic
The NIST e-Handbook, section 7.2.4.1, calls the normal-approximation interval p̂ ± z√(p̂(1 - p̂)/n) the expression "most frequently used," and then shows its weak spots: the lower limit can come out negative, which no proportion can be, and when the sample is small or the count of one outcome is very small, the symmetric limits "may not be accurate enough for some applications." NIST presents the Wilson interval as an alternative whose performance "does not strongly depend upon the value of n and/or p," and an exact binomial method for small counts.
In practice, the planning tables on this page are fine for setting a target. Where you should switch methods is the analysis of a small sample with a lopsided result, say 3 "yes" answers out of 40. The plus-or-minus from this formula would be misleading there, and a Wilson or exact interval is the right report. For small finite populations Penn State adds that, strictly, the count of successes follows a hypergeometric distribution, which gives an exact interval of its own. Our probability calculator handles the binomial building blocks if you want to see how lopsided small samples behave.
When 50% is too cautious
The p = 0.5 assumption is not a default for convenience. p(1 - p) peaks at 0.25 when p = 0.5 and falls off on both sides, so 0.5 gives the largest sample that any true proportion could need. If you have a reasonable prior estimate, from last year's survey or a pilot, the sample can be smaller.
| Expected proportion p | p(1 - p) | ±5%, large pop. | ±5%, N = 1,000 | ±3%, large pop. | ±3%, N = 1,000 |
|---|---|---|---|---|---|
| 50% | 0.2500 | 385 | 278 | 1,068 | 517 |
| 40% or 60% | 0.2400 | 369 | 270 | 1,025 | 507 |
| 30% or 70% | 0.2100 | 323 | 245 | 897 | 473 |
| 20% or 80% | 0.1600 | 246 | 198 | 683 | 407 |
| 10% or 90% | 0.0900 | 139 | 122 | 385 | 278 |
| 5% or 95% | 0.0475 | 73 | 69 | 203 | 169 |
Inputs: 95% confidence, z = 1.96, the same formulas, rounded up.
Penn State shows the trade-off with a presidential approval example. With an educated guess of 22% approval, a ±3% margin at 95% needs 0.22 × 0.78 × 1.96² / 0.03² = 732.47, so 733 people. The conservative p = 0.5 version needs 1,068. The course's rule for choosing between them is about cost: if going back to collect more data later would be expensive compared with the cost of each extra unit, take the conservative number. Its own exercise, testing CD player lifetimes from a shipment, picks the conservative route because sending a ship out again is too expensive.
Two cautions keep this honest. First, the guess may be wrong. Plan for 20% and observe 35%, and the margin you achieve is wider than the one you designed for. Second, most surveys ask many questions, and the sample has to serve the question closest to 50/50. A survey sized for one question expected near 10% will be underpowered for every other question on the form.
Completed responses, not invitations
Every count in these tables is the number of usable answers, not the number of people you contact. If you need 278 completed responses from a list of 1,000 and you expect about 30% to respond, the arithmetic says to invite 278 / 0.30 = 927 people, which is nearly the whole list. At a 20% response rate you would need 1,390 invitations, more people than the list contains, so the realistic options are a census of the full list with follow-up reminders or a looser margin.
Inviting more people fixes the count, not the bias. If the 30% who answer differ from the 70% who don't (busier staff skip the survey, unhappy customers answer it), the margin of error in these tables says nothing about that gap. The formulas measure only the random error of drawing some members instead of all of them.
The same limit applies to how the sample was chosen. The Penn State derivation starts from a simple random sample, where every member had an equal chance of selection, and every table on this page carries that assumption. An open web link shared on social media, a "whoever walks by" intercept, or an opt-in panel is not that, and quoting a ±5% margin for it overstates what you know.
Designs that sample in clusters, such as picking 10 classrooms and surveying every student in each, usually need a larger sample than these tables show, because students in one classroom tend to answer alike. Survey statisticians handle this with a design effect: the sample size from the table is multiplied by a factor that reflects how alike the clusters are. As a hypothetical illustration, a design effect of 1.5 would turn the 278 needed for a population of 1,000 into 417. Estimating the right factor for your design is a job for a sampling statistician, not a lookup table.
Subgroups shrink fast
A sample sized for the whole group gives a margin for the whole group only. Suppose 400 responses from a large population, ±4.9% overall. Split them evenly across four regions and each region has 100 responses, which carries about ±9.8%. Split them across two departments and each half of 200 carries about ±6.9%.
So the planning question is not "what margin do I want overall" but "what is the smallest group I intend to report separately, and what margin do I want for it." If a manager plans to compare the night shift (120 people) with the day shift (480 people) at ±5% each, use the table twice: 92 from the night shift (between the 100 and 200 rows; 1.96² × 0.25 / 0.0025 = 384.16, then 384.16 / (1 + 383.16 / 120) = 91.62) and 214 from the day shift (384.16 / (1 + 383.16 / 480) = 213.63). That is 306 responses in total, compared with 235 if you sized the 600-person workforce as one group at ±5%.
Differences between groups have their own, larger, error. If night shift satisfaction is 62% ± 5% and day shift is 55% ± 5%, the seven-point gap is not automatically "real" just because each number has a margin of five points. Testing a difference between two proportions is a hypothesis test with its own sample size calculation, and it adds a power term (the chance of detecting a real difference) that the estimation formula does not have. NIST section 7.2.4.2 shows the one-sample version of that calculation, testing a single proportion against a standard.
Which version of the formula is your software using?
If you check this page against another calculator and find a one-respondent difference, one of three choices is usually the cause.
The correction formula. Some sources use n₀ / (1 + n₀/N) instead of n₀ / (1 + (n₀ - 1)/N). The second is what falls out of Penn State's derivation with (N - 1) in the variance. At 95% and ±5% the two differ by less than a third of a respondent for every population we tested (50, 100, 200, 500, 1,000 and 10,000), and both round up to the same whole number: 45, 80, 132, 218, 278 and 370.
The z value. Some calculators use the more precise 1.959964 instead of 1.96. At ±5% that gives an unlimited-population n₀ of 384.15 instead of 384.16, and both round to 385. The difference grows only when the margin is very tight.
The rounding rule. Rounding to the nearest whole number instead of up turns 384.16 into 384, which falls just short of ±5%. Penn State rounds up, the tables here round up, and so should you.
The NIST critical values and Penn State's derivation agree with each other, so a cell on this page that disagrees with another tool by more than one respondent usually means the other tool is using a different p, a different z for the same confidence level, or no population correction at all.
Sizing a sample for an average instead of a percentage
Everything above is for a proportion: the share who said yes, the percentage defective, the fraction who prefer option A. For an average, such as mean hours worked or mean spending per visit, the variance is no longer fixed by p, and you have to supply a guess of the population standard deviation σ. Penn State's version for a mean with a finite population is:
where d is now the margin in the same units as the measurement. Penn State's worked example sizes a sample of plots out of 100 and gets 42.61, rounded to 43, then notes that ignoring the finite population correction would give 74.245, rounded to 75. In a population of 100 the correction nearly halves the sample, which is the same lesson the proportion table teaches in its top rows.
The course notes ask what the weak point of this formula is and answer that it is the variance estimate: "We do not know what this is!" Common sources are a pilot survey or the standard deviation from a previous round; if you have raw data from either, our statistics calculator gives the standard deviation to plug in. A guess that is too small gives a sample that is too small, so err high.
Picking your row in four steps
- Write down N, the number of people (or units) you could possibly sample. A mailing list of 1,800 is 1,800, not "the general public."
- Choose the margin for your smallest reported group, not for the whole survey. For most internal or association surveys ±5% is a reasonable target; ±3% is common for published polls; ±10% is a rough read on a small group.
- Pick the confidence level, 95% unless you have a reason. Read the cell, and remember it counts completed responses.
- Divide by your expected response rate to get the number of invitations, then decide whether that is achievable or whether a census of the whole group is simpler.
If your population or margin falls between the rows printed here, enter your own numbers into the sample size calculator with a 1,000-person population at ±5% and 95% preloaded, change any of the three fields, and it will return the exact count using the same formula as these tables. For more on why calculators round the way they do, the rounding and precision guide covers the general rules, and if you arrived here looking for profit margin rather than margin of error, the markup versus margin guide is the page you want. The rest of the math tools live on the math calculators page, and every long-form guide is listed on the guides hub.
Sources and inputs
- NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.7.1, Cumulative Distribution Function of the Standard Normal Distribution: critical values ±1.645, ±1.960, ±2.576. Checked October 30, 2026.
- NIST/SEMATECH e-Handbook of Statistical Methods, section 7.2.4.1, Confidence intervals for a proportion: normal approximation, Wilson interval and exact binomial method. Checked October 30, 2026.
- NIST/SEMATECH e-Handbook of Statistical Methods, section 7.2.4.2, Sample sizes required for testing a proportion. Checked October 30, 2026.
- Penn State Department of Statistics, STAT 506 Sampling Theory and Methods, Lesson 2: Confidence Intervals and Sample Size: finite-population variance of a sample proportion, sample size formulas for a proportion and a mean, conservative p = 0.5, and the worked examples cited above. Checked October 30, 2026.
Table inputs, repeated in one place: z = 1.645 (90%), 1.96 (95%), 2.576 (99%); p = 0.5 except in the expected-proportion table; finite population correction n = n₀ / (1 + (n₀ - 1)/N) applied to the unrounded n₀; sample sizes rounded up; margins of error rounded to one decimal place.