Margin of Error Calculator
The proportion your sample found, the margin of error around it, and the interval that follows — with the finite population correction when it applies.
Results
What this tool does
Every survey result is a estimate with a range around it, and the range is the part that gets left out of the headline. Put in how many people you asked and how many gave the answer you are counting, and this page returns the proportion, the margin of error at the confidence level you choose, and the interval the two produce. If you know the size of the whole population it applies the finite population correction, which matters a great deal when you have surveyed a club or a company and not at all when you have surveyed a country.
Formula
margin of error = z × √(p(1−p) ÷ n) · × the finite population correction when the sample is a large slice of the whole
Variables
| Symbol | Meaning | Unit |
|---|---|---|
sample_size | How many you asked | — |
successes | How many said yes | — |
confidence | Confidence level | % |
population | Size of the whole population, if you know it | — |
OUT | The proportion | % |
ME | Margin of error | % |
LO | Lower end of the interval | % |
UP | Upper end of the interval | % |
WD | The interval is this wide | % |
SE | Standard error | % |
ZZ | z for that confidence | — |
FP | Finite population correction applied | — |
FF | And it shrinks the error by | — |
N3 | For a 3-point margin you would need | — |
AS | The test's assumption holds | — |
Worked example
- How many you asked1000
- How many said yes520
- Confidence level95 %
- Size of the whole population, if you know it0
- The proportion52.00 %
- Margin of error3.10 %
- Lower end of the interval48.90 %
- Upper end of the interval55.10 %
- The interval is this wide6.19 %
- Standard error1.580 %
- z for that confidence1.9600
- Finite population correction appliedNo
- And it shrinks the error by1.0000
- For a 3-point margin you would need1066
- The test's assumption holdsYes
Limitations
- The calculation runs entirely in your browser. The values you type are never sent to a server.
- The result is an estimate based only on the values you type. Real situations often include factors this calculator does not know about.
Frequently asked questions
What does a margin of error of three points mean?
That if you repeated the same survey many times, drawing a fresh random sample each time, about ninety-five of every hundred intervals built this way would contain the true figure. It says nothing about whether this particular interval does. It also covers only the error from sampling — it says nothing about people who refused to answer, questions that were worded to lead, or a sample that was never random to begin with. Those are usually the larger problem and no arithmetic fixes them.
When does the population size matter?
Only when your sample is a large fraction of it — above roughly five per cent. The famous fact that a thousand people can represent a whole country is true precisely because at that ratio the population size drops out of the arithmetic almost entirely. But if you ask 200 of the 400 members of a club, half the population is in your sample and there is far less left to be uncertain about; the finite population correction shrinks the margin accordingly. Leave the box at zero and no correction is applied.