Sample Size Calculator

n₀ = z² · p (1 − p) / e² small population: n = n₀ / ( 1 + (n₀ − 1) / N )

z

The critical value for the confidence level — 1.9600 at 95%, 1.6449 at 90% and 2.5758 at 99%. It is printed in its own row because it is the only number in the formula that comes from the confidence level rather than from your assumptions about the population

p

The proportion you expect to find, as a decimal. This is the one input that is a guess, and it enters as p(1 − p), which is largest at p = 0.5 — so entering 50% is the assumption that produces the biggest sample and the safest one

e

The margin of error, as a decimal — the half-width you are willing to accept, in the same units as the proportion. It is squared in the denominator, which is why halving it roughly quadruples the sample

n₀

The sample size for a population large enough that the population size does not matter. It is the answer whenever the population is unknown, and it is the input to the correction rather than the final answer when it is not

N

The size of the population, entered as 0 when it is unknown or effectively unlimited. The correction divides n₀ by a number larger than the fraction n₀/N, so the corrected sample always comes out below N — asking more people than exist is not a risk this formula can produce

Use it before collecting data, whenever the cost of the sample is real — a survey with an incentive per response, a lab measurement with a per-sample price, a clinical study with recruitment targets. The order of the decisions is: fix the margin of error you can live with, fix the confidence level, then guess the proportion, then read the sample size. Only the first of those is a judgement about the study; the other two are conventions or guesses. If the answer is unaffordable, the honest move is to widen the margin of error and say so in the report, because a margin of error you cannot fund is not a target. Enter the population size whenever the population is genuinely small — under a few thousand — because the correction is substantial there and ignoring it means recruiting far more people than the study needs.

1. A national survey at 95% confidence and ±5 points

   1. Critical value at 95% confidence: 1.96
   2. With p = 0.5 the product p(1 − p) is 0.25, its largest possible value
   3. n₀ = 1.96² × 0.25 / 0.05² = 3.8416 × 0.25 / 0.0025 = 384.1459
   4. Rounded up to a whole person: 385 responses

   385 is the number behind the familiar claim that a poll of about a thousand people is accurate to within three points — tighten the margin to ±3 and the answer becomes 1068, which is where that figure comes from. The population size was left at 0 because a national survey is a negligible fraction of the country, and at that ratio the correction would change the answer by less than one person.

2. The same survey with a tighter ±3 point margin

   1. Critical value unchanged at 1.96, because the confidence level did not change
   2. n₀ = 1.96² × 0.25 / 0.03² = 0.9604 / 0.0009 = 1067.0719
   3. Rounded up: 1068 responses
   4. Compare with 385: the margin shrank by a factor of 1.667 and the sample grew by a factor of 2.78, which is 1.667²

   Precision is bought quadratically. Moving from ±5 to ±3 points is not a 67% increase in work but a 178% increase, because the margin of error is squared in the denominator. The practical consequence is that there is a floor on how precise a survey can cheaply be, and that advertising a margin tighter than about ±3 points means a sample in the thousands — which is exactly the range where non-response bias starts to matter more than the sampling error the formula is controlling.

3. A staff survey of a company with 1000 employees

   1. The uncorrected figure is still 384.1459, since it does not know about the population yet
   2. Correction: 384.1459 / ( 1 + 383.1459 / 1000 ) = 384.1459 / 1.3831459 = 277.73
   3. Rounded up: 278 employees
   4. That is 28% of the workforce rather than the 38% the uncorrected figure would have implied

   The correction is what makes small-population surveys affordable: 278 rather than 385, a saving of more than a hundred interviews for the same stated margin. It grows as the sample becomes a larger share of the population. At 100 employees the answer is 80 — you cannot sample more than you have — and at a million employees the correction is worth a single person, which is why 0 for an unknown population is a reasonable default rather than a shortcut.

The formula sizes a sample of responses, not a list of invitations. If a third of the people asked do not reply, 385 responses means asking about 580, and if the people who reply differ systematically from those who do not, no sample size fixes the resulting bias — it only makes the sampling error smaller than the total error. The model also assumes each response is an independent draw from the population, which fails for clustered designs: a survey of 400 students drawn from 8 classrooms has an effective sample much closer to 8, and the count from this page should be multiplied by a design effect before it is used. The estimate is for a single proportion; a mean requires a standard deviation instead, which this page does not take, and a survey that will be broken into subgroups needs a sample sized for the smallest subgroup rather than for the whole. Finally, all three confidence levels offered are conventions, and none of them accounts for the several comparisons a real analysis usually makes.

How many survey responses do I need?

For the conventional ±5 percentage points at 95% confidence, 385 — and 1068 if you want ±3 points. Those two numbers cover the large majority of published surveys, and they assume a population large enough that its size is irrelevant. If the population is small enough that you could plausibly survey a large share of it, enter the size and the correction will lower the figure: 1000 employees need 278 responses rather than 385, and 200 employees need 132. Anchor on the margin of error first, because that is the number your audience will actually read.

Why does the expected proportion default to 50%?

Because p(1 − p) is largest at p = 0.5, and the sample size is proportional to it, so entering 50% gives the largest answer any proportion could require. That makes it the safe assumption when you genuinely do not know: a survey that comes back at 60% will have a margin of error slightly better than the one you planned. Being more specific pays off — at p = 90% the term falls to 0.09 from 0.25, so the same margin of error needs 36% of the sample. Only use a value other than 50% if you have a defensible prior, such as a previous wave of the same survey.

What does the finite population correction do?

It shrinks the sample when the population is small enough that drawing from it without replacement measurably reduces the uncertainty. The effect depends on what share of the population you are sampling: at 10% the correction is worth a few per cent of the sample, at 28% it is worth about a quarter, and at 50% it is worth nearly a third. Enter 0 for the population size when it is unknown or very large, and enter the real figure when the population is under a few thousand — this is the field that most often changes an answer people had already calculated.

Which margin of error and confidence level should I aim for?

±5 points at 95% is the default in most published work, ±3 points where a decision hinges on a close result, and ±10 points for exploratory work where only a large difference matters. On the confidence level, 95% is the convention and 99% costs about 73% more sample for an interval only a third wider. Before tightening either, check what the answer costs: ±3 points at 99% confidence needs 1843 responses, roughly five times the default. If that is unaffordable, report the wider margin rather than recruiting a sample you cannot complete.

Why does the answer round up to a whole person?

Because the fractional figure is the exact arithmetic and a whole number is what can be recruited, and the two conventions for turning one into the other point in different directions. Rounding to the nearest could round 384.1459 down to 384, which misses the target margin of error by a hair — and the target is the thing you asked for. Rounding up guarantees the margin is met or beaten, costs at most one extra interview, and matches how sample sizes are specified in a protocol, where 385 is a commitment and 384.1459 is an intermediate value.

Why is there no table of sample sizes on this page?

Because the table that people want here is a grid of confidence level against margin of error, and that grid is what the panel already computes. A table printed at a fixed 95% would contradict the panel the moment you switched to 99%, and having the page disagree with itself is worse than having no table — a reader would have to decide which of the two numbers to believe. The second reason is that the table would have to pick a proportion too, and 50% is only the right choice while it remains the safe default. Change confidence level, margin of error, proportion or population size and the answer recomputes, which is the whole content of the table people are looking for.

• 3.1 Terminology — Introductory Statistics 2e (relative frequency approaching probability as a sample grows, which is the property that makes a target margin of error meaningful in advance) — OpenStax (Rice University)
• 1.3.6.1. What is a Probability Distribution — e-Handbook of Statistical Methods (the distribution of an estimate, which is what the margin of error is a quantile of) — National Institute of Standards and Technology (NIST)
• 1.3.6.7.1. Cumulative Distribution Function of the Standard Normal Distribution — e-Handbook of Statistical Methods (the cumulative function the critical values come from; the same values, used in reverse here to size a sample rather than to bound an estimate) — National Institute of Standards and Technology (NIST)