Your survey
How often the interval should contain the true value across repeated surveys. 95 is the working standard.
How wide a band around the result you are willing to accept.
The split you expect. Leave at 50 unless you have prior data — it is the most demanding case.
The total group you are drawing from. Blank means large enough that it makes no difference.
Completed responses needed
385
At 95% confidence with a ±5.0% margin of error.
Critical value (z)
1.9600
Before population correction
384.15
No correction applied
Variability term p(1 − p)
0.2500
Maximum is 0.25 at a 50/50 split
Step-by-step solution for your numbers
Each row shows the formula, your own figures substituted into it, and the value that comes out — so you can reproduce the answer by hand and defend it to whoever asks.
Step 1: Find the critical value for the confidence level
- Formula
- z = z_(α/2) where α = 1 − confidence
- Your numbers
- α = 1 − 95.00% = 0.0500, so z = z_(0.0250)
- Result
- z = 1.959964
The two-sided critical value cuts α/2 of the standard normal distribution into each tail.
Step 2: Convert the percentages to decimals
- Formula
- p = response distribution ÷ 100, e = margin of error ÷ 100
- Your numbers
- p = 50.00 ÷ 100, e = 5.00 ÷ 100
- Result
- p = 0.5000, e = 0.0500
Step 3: Compute the variability term
- Formula
- p(1 − p)
- Your numbers
- 0.5000 × (1 − 0.5000) = 0.5000 × 0.5000
- Result
- 0.250000
This term peaks at 0.25 when p = 0.5 — maximum uncertainty, and therefore the largest sample requirement.
Step 4: Apply Cochran's formula
- Formula
- n₀ = z² · p(1 − p) / e²
- Your numbers
- n₀ = 1.959964² × 0.250000 ÷ 0.0500² = 3.841459 × 0.250000 ÷ 0.002500
- Result
- n₀ = 384.1459
This is the requirement for a population large enough that removing your sample from it changes nothing.
Step 5: Round up to a whole respondent
- Formula
- n_required = ⌈n⌉
- Your numbers
- ⌈384.1459⌉
- Result
- 385 completed responses
Always round up: rounding down would leave the interval wider than the one you specified.
The formula
n₀ = z² · p(1 − p) / e²
n = n₀ / (1 + (n₀ − 1)/N)
z is the two-sided critical value for your confidence level, p the assumed response proportion, e the margin of error as a decimal, and N the population size. The second line is the finite-population correction, applied only when N is known.
What each input actually controls
Confidence level sets how often the interval would contain the true population value if the survey were repeated many times over. It enters the formula as z, and it enters squared — so moving from 95% to 99% multiplies the required sample by about 1.73, not by a few percent.
Margin of error is the half-width of the interval around your result. It appears in the denominator squared, which makes it the most expensive input on the page: halving the margin quadruples the sample. Going from ±5% to ±1% at 95% confidence takes the requirement from 385 to 16,588.
Response distribution is the split you expect to find. The term p(1 − p) peaks at 0.25 when p is 0.5, so a 50/50 assumption always produces the largest requirement. That is why it is the default: it is the assumption you cannot be caught out by.
Population size matters far less than people expect. It only bites once your sample is a meaningful fraction of the whole group — surveying 385 of 500 employees is a different exercise from surveying 385 of a million, and the correction is what tells the two apart.
This page answers “how many responses do I need?”. Once the survey is in and you want the precision your actual sample achieved — the reverse of this calculation — use the margin of error calculator
Methodology and sources
This calculator implements Cochran's sample size formula for estimating a population proportion, with the standard finite-population correction applied when a population size is supplied. Critical values are not read from a lookup table: z is computed directly by inverting the complementary error function, so any confidence level between 0 and 100 is exact rather than interpolated between table rows.
Why this page says 1,068 where most published tables say 1,067.For ±3% at 95% confidence, Cochran's formula gives n₀ = 1067.07. At n = 1067 the achieved margin is 3.0001% — marginally wider than requested — so this calculator rounds up, as the formula requires, rather than rounding to the nearest whole number the way most printed tables do. The difference is one respondent and both are defensible; ours is simply the conservative reading of “at most ±3%”.
All arithmetic runs in your browser. Nothing you type is transmitted, stored, or logged — the page makes no network request to produce an answer.
Two assumptions are worth stating plainly, because no calculator can check them for you: the formula assumes simple random sampling from the population you care about, and it accounts only for sampling error. Non-response, coverage gaps and question wording are usually the larger sources of error in a real survey, and none of them appear in this arithmetic. Federal statistical agencies document how they handle each of these; the sources below are the working references.
Primary sources
U.S. Census Bureau
Survey methodology and sampling documentationFederal survey programs publish their sample designs, response rates and variance estimation methods — the reference standard for how a probability sample is specified in practice.
U.S. Census Bureau
American Community Survey — sample size and data qualityDocuments how sample size, margins of error and response rates interact at production scale, including the published margins that accompany every ACS estimate.
U.S. Bureau of Labor Statistics
Handbook of MethodsChapter-level documentation of the sampling and estimation methods behind federal labor statistics, including how sampling error is calculated and reported.
Links go to the publishing agency, so you always read the current figure rather than a copy of it. This page embeds no agency data of its own.
Common questions
Why is 385 the answer so often?
Because 95% confidence, a ±5% margin of error and a 50/50 response split are the most common defaults, and those inputs give n₀ = 384.15, which rounds up to 385. The figure barely moves once the population is above roughly 20,000, which is why so many surveys land on it.
Do I need a bigger sample if my population is bigger?
Almost never. Sample size depends on the variability you are measuring, not on the size of the group you draw from. Going from a population of 100,000 to 100 million changes the requirement at 95%/±5% by less than one respondent. Population size only matters when the sample is a large share of it, which is what the finite-population correction handles.
What should I use for the response distribution?
Use 50% unless you have real prior data. p(1 − p) is largest at 0.5, so 50% produces the largest — and therefore safest — required sample. Entering a lopsided split shrinks the requirement, but only justifiably if you already know the split.
Is this the sample size I need, or the number of people I need to invite?
It is the number of completed responses. Invitations must be scaled up by your expected response rate: at a 20% response rate, 385 completions needs roughly 1,925 invitations. Nothing on this page models non-response bias, which is usually a larger threat to a survey than sampling error.
Why can’t I enter 100% confidence?
There is no finite critical value at 100%. Total certainty about a population would require surveying every member of it — a census. The calculator refuses the input rather than silently substituting a large number.
Does a bigger sample fix a bad sample?
No. These formulas assume a random sample from the population of interest. If the people who respond differ systematically from those who do not, a larger sample makes the wrong answer more precise, not more correct.
This page does the arithmetic. We check the numbers you put into it.
Sizing the survey is the easy half — the hard half is knowing whether the numbers you are benchmarking against hold up. A $29 Verified Snapshot answers one business question in a short, fully cited brief — and every claim in it is checked against its own source before it ships. Others check that links work. We check that claims are true.
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These calculators are provided for general business analysis and are independent informational research — not investment, financial, legal, or tax advice. The arithmetic is standard and openly documented on this page; the assumptions you enter are yours, and the conclusions drawn from them are too.