Your result
The number of completed responses you actually collected.
The proportion you measured — the share who answered one way.
95 is the reporting standard for published survey results.
Only narrows the interval when your sample is a large share of the whole group.
Margin of error
±3.10%
Your result of 50.0% carries a 95% confidence interval of 46.90% to 53.10%.
Confidence interval
46.9% – 53.1%
Standard error
0.01581
Critical value (z)
1.9600
No population correction
Step-by-step solution for your numbers
Every row uses the figures you entered, so the arithmetic can be checked line by line rather than taken on trust.
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
- Result
- z = 1.959964
Step 2: Convert the response distribution to a proportion
- Formula
- p = response distribution ÷ 100
- Your numbers
- p = 50.00 ÷ 100
- Result
- p = 0.5000
Step 3: Compute the standard error of the proportion
- Formula
- SE = √( p(1 − p) / n )
- Your numbers
- SE = √( 0.5000 × 0.5000 ÷ 1,000 ) = √( 0.00025000 )
- Result
- SE = 0.015811
The standard error shrinks with the square root of n — quadrupling the sample halves it.
Step 4: Multiply by the critical value
- Formula
- MOE = z · SE
- Your numbers
- MOE = 1.959964 × 0.015811
- Result
- MOE = 0.030990
Step 5: Express the result in percentage points
- Formula
- MOE% = MOE × 100
- Your numbers
- 0.030990 × 100
- Result
- ±3.10%
At 95% confidence, a result of 50.0% means the true value is between 46.90% and 53.10%.
The formula
MOE = z · √( p(1 − p) / n )
FPC = √( (N − n) / (N − 1) )
z is the two-sided critical value for your confidence level, p the observed proportion, n the sample size, and N the population size. The correction factor is applied only when N is known, and it multiplies the margin.
Reading the number properly
The margin of error answers one narrow question: how much the result could move purely because of which people happened to land in your sample. It is a statement about the sampling procedure, and it is silent on everything else that can go wrong in a survey.
Two consequences follow, and both are routinely missed. First, precision improves with the square root of the sample — quadrupling your responses only halves the margin, which is why surveys tend to settle around a thousand responses rather than pushing higher. Second, the margin is not constant across a questionnaire: it is widest at a 50/50 split and narrows toward the extremes, so the single headline figure a report quotes is normally the worst case rather than the figure attaching to any particular question.
When you are comparing two numbers rather than reporting one, this is not the right tool. Overlapping intervals do not settle whether two rates differ — test the difference directly with the A/B significance calculator instead.
This page tells you the precision a sample you already have achieved. To work the other way — how many responses you need to hit a target margin before you field the survey — use the survey sample size calculator
Methodology and sources
This page computes the normal-approximation (Wald) interval for a single proportion, with the standard finite-population correction applied when a population size is supplied. Critical values are computed exactly by inverting the complementary error function rather than being read off a table, so any confidence level is handled precisely.
Where the normal approximation is unreliable — fewer than about five expected responses in either category, or a result at 0% or 100% — the page says so beside the answer rather than presenting a confident interval it does not deserve. For results near the boundaries, an exact method such as the Clopper-Pearson interval is the appropriate substitute.
All arithmetic runs in your browser; nothing you enter is transmitted or stored. The federal survey programs below are the working references for how sampling error is estimated and disclosed in practice, including on complex sample designs where this simple formula is only a starting point.
Primary sources
U.S. Census Bureau
American Community Survey — margins of error and how to use themEvery ACS estimate is published with its own margin of error at 90% confidence; the guidance explains how the agency computes and expects users to interpret them.
U.S. Bureau of Labor Statistics
Handbook of Methods — reliability of estimatesDocuments how sampling error is estimated and reported across federal labor statistics, including the distinction between sampling and non-sampling error.
U.S. Census Bureau
Current Population Survey — source and accuracy statementsWorking examples of variance estimation on a complex sample design, where a simple random sample formula is a starting point rather than the final answer.
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
What does a ±3% margin of error actually mean?
It describes the method, not the single result in front of you. If the same survey were repeated many times with fresh random samples, about 95% of the intervals built this way would contain the true population value. It does not mean there is a 95% chance that this particular interval contains it — that is a different, and much stronger, claim.
Why do two results in the same survey have different margins of error?
Because the margin depends on the proportion being measured. A 50/50 split has the widest interval; a result near 5% or 95% has a considerably narrower one. Published surveys usually quote the widest case, computed at 50%, as a single headline figure covering the whole instrument.
Can two results be inside each other’s margin of error and still differ significantly?
Yes, and the reverse also happens. Comparing two overlapping intervals is not the same test as comparing the difference between them directly. For a head-to-head comparison of two rates, use a two-proportion test rather than eyeballing whether the intervals touch.
What is the finite-population correction and when does it apply?
It is a factor of √((N − n)/(N − 1)) that narrows the interval once your sample is a meaningful share of the population. Surveying 500 people out of 1,000 genuinely gives more precision than surveying 500 out of a million. At a full census it drives the sampling error to zero, because there is nothing left to sample.
Does the margin of error cover every kind of error in a survey?
No. It covers sampling error only — the luck of which people ended up in your sample. Non-response, coverage gaps, question wording and interviewer effects are not in this formula and are frequently larger. A tight margin of error on a badly drawn sample is precision about the wrong thing.
Why does the interval get clipped at 0% and 100%?
Because a proportion cannot fall outside that range. When a result is close to either boundary, the symmetric normal-approximation interval runs past it, which is a sign that the approximation is straining — exact methods such as the Clopper-Pearson interval are more appropriate near the edges.
This page does the arithmetic. We check the numbers you put into it.
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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.