Survey sample size calculator
Find out how many completed responses a survey needs. Set the confidence level, margin of error and expected proportion, add the population size if it is small and known, and get the answer with the working shown.
For 95% confidence and a margin of error of ±5 percentage points, a survey needs 385 completed responses when the population is large and the likely answer split is unknown (use 50%). If the whole population is only 1,000 people, the finite population correction brings that down to 278.
Required sample size
385 completed responses
To estimate a percentage within ±5 points at 95% confidence, when about 50% of people give the answer you are measuring.
- Confidence level
- 95% (z = 1.960)
- Margin of error
- ±5 percentage points
- Expected proportion
- 50% (the most cautious choice)
- Population
- Not set (treated as very large)
Workingn₀ = z² × p × (1 − p) ÷ e² = 1.960² × 0.5 × 0.5 ÷ 0.05² = 384.15Round up to a whole respondent: 385
How does the sample size calculator work?
It uses Cochran's formula for estimating a single proportion from a simple random sample. Four inputs decide the result:
- Confidence level. How sure you want to be that the interval around your result contains the true value for the whole population. It sets the critical value z: 1.645 at 90%, 1.960 at 95% and 2.576 at 99%. The calculator computes z from the normal distribution, so any level works.
- Margin of error. The half-width of that interval, in percentage points. With a margin of ±5, a result of 40% is reported as 35% to 45%.
- Expected proportion. The share of people you expect to give the answer you are measuring. The required sample is largest at 50%, so 50% is the cautious default when you have no estimate.
- Population size (optional). Only matters when the population is small and known. It applies the finite population correction.
n₀ = z² × p × (1 − p) ÷ e²n = n₀ ÷ (1 + (n₀ − 1) ÷ N) (only when the population N is known)invitations = n ÷ response rate
Here z is the two-sided critical value of the standard normal distribution, p is the expected proportion, e is the margin of error as a fraction (0.05 for ±5 points) and N is the population size. Round up only at the end: rounding n₀ up to 385 first and then correcting would give 279 instead of 278 for a population of 1,000.
The correction shrinks the sample because sampling without replacement from a small population tells you more per response than sampling from an unlimited one. It is usually negligible when the sample is below about 5% of the population.
What is a worked example?
A company wants to survey its 1,000 customers with 95% confidence and a margin of error of ±5 points, and expects the answer split to be unknown, so p = 0.5. Here z = 1.960.
n₀ = 1.960² × 0.5 × 0.5 ÷ 0.05² = 3.8415 × 0.25 ÷ 0.0025 = 384.15n = 384.15 ÷ (1 + (384.15 − 1) ÷ 1,000) = 277.73Round up: 278 completed responses
Without the population correction the answer would be 385. If the company expects a 25% response rate, it should invite about 1,112 customers (278 ÷ 0.25).
If earlier research shows that only about 20% of people give the answer, setting p = 0.2 lowers the requirement for a large population to 246. Using 50% is the safe choice because it can only overstate the sample needed.
What sample sizes do common settings give?
These tables use the same formula as the calculator, with an expected proportion of 50%.
| Margin of error | 90% confidence | 95% confidence | 99% confidence |
|---|---|---|---|
| ±1% | 6,764 | 9,604 | 16,588 |
| ±2% | 1,691 | 2,401 | 4,147 |
| ±3% | 752 | 1,068 | 1,844 |
| ±4% | 423 | 601 | 1,037 |
| ±5% | 271 | 385 | 664 |
| ±7.000000000000001% | 139 | 196 | 339 |
| ±10% | 68 | 97 | 166 |
Halving the margin of error roughly quadruples the sample, because the margin shrinks with the square root of the sample size.
| Population | Sample needed | Share of population |
|---|---|---|
| 50 | 45 | 90.0% |
| 100 | 80 | 80.0% |
| 250 | 152 | 60.8% |
| 500 | 218 | 43.6% |
| 1,000 | 278 | 27.8% |
| 2,500 | 334 | 13.4% |
| 5,000 | 357 | 7.1% |
| 10,000 | 370 | 3.7% |
| 100,000 | 383 | 0.4% |
| 1,000,000 | 384 | 0.0% |
Above roughly 100,000 people the population barely matters: 383 for 100,000 against 385 for an unlimited population. A tighter margin of ±3 points needs 1,068 responses.
When should you not use this calculator?
The formula answers one question: how large must a simple random sample be to estimate one percentage to a given precision? It is the wrong tool when:
- The sample is not random. Opt-in online panels, social media polls and convenience samples have no true sampling margin of error, however large they are. Survey bodies such as AAPOR advise against reporting a margin of sampling error for non-probability samples.
- The design is complex. Stratified, clustered or heavily weighted samples usually need a larger sample. Multiply the result by the design effect (deff), which is 1 for a simple random sample and above 1 when weighting or clustering reduces precision.
- You will compare groups. Detecting a difference between two groups or waves needs a power calculation, which also depends on the size of difference you care about. This calculator sizes a single estimate.
- You will report subgroups. The result is for the total sample. Each subgroup you want to report on at the same precision needs its own sample of this size.
- You are estimating an average. For means, such as an average rating, the formula uses the variance of the measure instead of p × (1 − p).
- Bias is the bigger risk. Non-response, coverage gaps and poor question wording are not fixed by a bigger sample.
Frequently asked questions
What sample size do I need for a survey?
It depends on how precise you need to be. For 95% confidence and a margin of error of ±5 points, about 385 completed responses from a large population; ±3 points needs 1,068. For a small, known population, use the population size field to reduce the number.
Does population size matter?
Only when the sample is a sizeable share of the population, roughly more than 5% of it. For a population of 1,000 the requirement falls from 385 to 278; for 100,000 it is 383, almost the same as for an unlimited population.
Why use 50% as the expected proportion?
The product p × (1 − p) is largest at 50%, so 50% gives the biggest sample for the same precision. If you do not know the likely result, 50% guarantees the margin of error will be no wider than you asked for.
How many people should I invite?
Divide the required completes by the share of invited people you expect to respond. With 278 completes needed and a 25% response rate, invite about 1,112. Enter the response rate above to have the calculator do this.
Is the sample size enough for subgroups?
No. The result applies to the total sample. If you want to report a segment, such as customers in one region, that segment needs enough responses by itself, so the total must be larger. Work out the sample the segment needs, then divide by the segment's share of the population to find the total.
What is the difference between margin of error and confidence level?
The margin of error is how wide the interval is; the confidence level is how often intervals built this way would contain the true value. Raising the confidence level or shrinking the margin both increase the sample needed. The margin of error calculator works in the other direction.
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Formulas and references
The standard sources behind the formulas on this page:
- Cochran, W. G. (1977). Sampling Techniques (3rd ed.). New York: John Wiley and Sons. Sample size for a proportion and the finite population correction.
- Kish, L. (1965). Survey Sampling. New York: John Wiley and Sons. Design effects.
- Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions. Normal distribution functions (series 26.2.10 and the continued fraction 26.2.14, used to compute the normal distribution).
- Acklam, P. J. (2003). An algorithm for computing the inverse normal cumulative distribution function, with one step of Halley's method (used to find the critical value z).
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