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Free margin of error calculator

Margin of error calculator: how precise are your results?

Enter how many people answered your survey and pick a confidence level. You get the margin of error in percentage points, with the formula, a worked example and what it does and does not cover.

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How do you calculate the margin of error?

Multiply the z-score for your confidence level by the square root of p × (1 − p) ÷ n, where n is your number of responses and p the observed proportion. With 400 responses at 95% confidence and p = 50%, the margin of error is ±4.90%. Claform’s free calculator also corrects for small populations.

Calculator

Margin of error calculator: how precise are your results?

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Number of completed responses.
95% is the usual choice for surveys.
Leave empty if the population is very large or unknown.
Share who gave the answer you report. 50% is the worst case.

Margin of error

±4.9%

Assumes a simple random sample. It covers sampling error only, not non-response or question bias.

Formula

The margin of error formula

For a proportion from a simple random sample, the margin of error is the z-score times the standard error. When the sample is a large share of the population, a finite population correction makes it smaller.

MOE = z × √(p × (1 − p) ÷ n) × √((N − n) ÷ (N − 1))

  1. Pick a confidence level and its z-score: 80% = 1.2816, 85% = 1.4395, 90% = 1.6449, 95% = 1.96, 99% = 2.5758.
  2. Take p, the proportion you are reporting, as a decimal. Use 0.5 if you want the widest, most cautious margin.
  3. Calculate the standard error: √(p × (1 − p) ÷ n), where n is the number of completed responses.
  4. Multiply by z. This is the margin of error for a very large population.
  5. If you know the population size N, multiply by √((N − n) ÷ (N − 1)). Leave this step out when the population is very large.
  6. Multiply by 100 to express it in percentage points, for example 0.049 becomes ±4.90%.

Worked example

Worked example: 400 responses at 95% confidence

You surveyed 400 people at random from a large customer base, and 50% said they would buy again. Use z = 1.96 and p = 0.5.

Standard error = √(0.5 × 0.5 ÷ 400) = √0.000625 = 0.025. Margin of error = 1.96 × 0.025 = 0.049, or ±4.90%.

So the true share is likely between 45.1% and 54.9%. “Likely” here means 95% confidence: if you repeated the survey many times, about 95 out of 100 such intervals would contain the true value.

If your whole customer base is only 2,000 people, the correction √(1,600 ÷ 1,999) ≈ 0.895 narrows the margin to ±4.38%.

Sample size drives precision, but with diminishing returns: 100 responses give ±9.80%, 400 give ±4.90% and 1,600 give ±2.45%. Halving the margin takes four times the responses.

How to read a margin of error

The margin of error tells you how much a result could move from sampling alone.

  • Read results as a range

    A result of 50% ± 4.9% means the true value is likely between 45.1% and 54.9%. Report the range, not just the single number.

  • Close results may be a tie

    If two options are closer together than the margin of error, the data does not clearly show which one is ahead.

  • Subgroups are less precise

    A segment of your sample has fewer responses, so its margin of error is wider. Calculate it again with the segment’s own count.

  • Sampling error is not the only error

    The margin ignores non-response, leading or unclear wording and non-random samples. A small margin cannot fix a biased survey.

Margin of error questions

What margin of error is acceptable for a survey?

Many surveys aim for ±5% or less at 95% confidence. For a quick internal check, ±10% may be enough. For decisions where a few points matter, aim for ±3% or tighter. Choose the margin that would still lead you to the same decision at either end of the range.

What does a 95% confidence level mean?

If you ran the same survey many times, each time with a new random sample, about 95 out of 100 of the ranges you calculate would contain the true value. It does not mean there is a 95% chance your particular result is exactly right.

Why use 50% as the proportion?

p × (1 − p) is largest at 50%, so it gives the widest margin. Using 50% is the cautious choice when you report many questions with different results. For a single result far from 50%, enter that value for a tighter, more exact margin.

Does population size affect the margin of error?

Only when your sample is a noticeable share of the population. For a large population, 400 responses give about ±4.90% at 95% confidence whether the population is 100,000 or 10 million. For a small population, such as 2,000 employees, the finite population correction makes the margin smaller.

How do I reduce my margin of error?

Collect more completed responses. Because the margin shrinks with the square root of the sample size, you need four times the responses to halve it. Lowering the confidence level also narrows the margin, but your results become less certain.

Does the margin of error include all survey errors?

No. It only measures random sampling error. It does not cover people who did not respond, biased or confusing questions, data entry mistakes, or a sample that was not chosen at random.

Is this margin of error calculator free?

Yes. It is free, needs no sign-up and runs in your browser. Nothing you enter is sent or stored.

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