Margin of error calculator

How does the margin of error calculator work?

The margin of error is the half-width of a confidence interval for a percentage. It is computed from the standard error of a proportion, scaled by the critical value z for the confidence level:

e = z × √(p × (1 − p) ÷ n)e = z × √(p × (1 − p) ÷ n) × √((N − n) ÷ (N − 1)) (finite population of size N)

Here n is the number of completed responses, p is the proportion (as a fraction) and z is 1.645, 1.960 or 2.576 at 90%, 95% or 99% confidence. The interval for the whole population is the result plus or minus e. Because p × (1 − p) is largest at 50%, a margin of error quoted for a whole survey is normally the one at 50%, the most cautious case.

The second formula applies the finite population correction, which matters when the sample is a sizeable share of a small population (roughly more than 5%). When the sample is the whole population there is no sampling error at all.

What is a worked example?

A survey collects 1,000 responses and a result of 50% at 95% confidence.

standard error = √(0.5 × 0.5 ÷ 1,000) = 0.01581e = 1.960 × 0.01581 = 0.0310, or ±3.10 points

So a reported 50% is likely to be within about 3 points of the true figure for the whole population. If a different result of 40% is found with the same sample, the margin is slightly narrower, ±3.04 points, giving 37.0% to 43.0%.

With a population correction, 278 responses from a population of 1,000 give a margin of ±4.997 points, just under 5, which matches the result of the sample size calculator.

What margin of error does a given sample give?

Margins of error, in percentage points, for a result of 50% from a large population:

Each extra response helps less than the last. Going from 1,000 to 2,000 responses narrows the margin by about 0.9 points; going from 100 to 1,000 narrows it by about 6.7.

When should you not use this calculator?

A margin of error covers sampling error only, and only for a random sample. Do not rely on it when:

• The sample is not random. Opt-in panels and convenience samples have no true sampling margin of error. Some pollsters publish a credibility interval instead, which rests on modelling assumptions. AAPOR advises against quoting a margin of sampling error for non-probability samples.
• Other errors dominate. Coverage gaps, non-response, question wording and weighting add error that the margin of error does not include.
• The data are weighted or clustered. Weighting and clustering widen the real margin. Divide the sample size by the design effect to get an effective sample size, and use that as n.
• You are comparing two results. The margin for the difference between two independent results is larger than for either alone, about 1.4 times larger when the two samples are similar in size. Use the survey significance calculator.
• The sample or the proportion is extreme. For small samples or percentages near 0% or 100%, this normal-approximation interval is unreliable, and a Wilson or exact interval is better.

Frequently asked questions

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Related tools

• Survey sample size calculatorHow many completed responses do you need? Set the confidence level, margin of error, expected proportion and, if you know it, the population.
• Survey significance calculatorIs the gap between two segments or waves real or noise? A two-proportion z-test with the difference, z, p-value and a plain-English reading.
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Formulas and references

The standard sources behind the formulas on this page:

1. Cochran, W. G. (1977). Sampling Techniques (3rd ed.). New York: John Wiley and Sons. Standard error of a proportion and the finite population correction.
2. Kish, L. (1965). Survey Sampling. New York: John Wiley and Sons. Design effects and effective sample size.
3. Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions. Normal distribution functions (series 26.2.10 and the continued fraction 26.2.14, used to compute the normal distribution).
4. Acklam, P. J. (2003). An algorithm for computing the inverse normal cumulative distribution function, with one step of Halley's method (used to find the critical value z).