Margin of Error Calculator

The margin of error is the half-width of a confidence interval: the distance from the sample estimate to each end of the interval. A poll that finds 52% support with a margin of error of ±3.1 percentage points at 95% confidence says that the support in the whole population is plausibly between 48.9% and 55.1%. It measures one thing only, sampling error: the chance variation that comes from asking a random sample instead of everyone.

The size of the margin depends on the variation in the answers, on the sample size and on the confidence you ask for. Confidence 95% means that if the survey were repeated many times with new random samples, about 95% of the intervals built this way would contain the true value. It does not mean that this one interval has a 95% probability of containing it.

Here p is the sample proportion, n the sample size, σ the population standard deviation and s the sample standard deviation. The multiplier z depends only on the confidence level: 1.2816 for 80%, 1.6449 for 90%, 1.9600 for 95%, 2.5758 for 99% and 3.2905 for 99.9%. The t multiplier is larger, most of all for small samples, because s is itself an estimate. This calculator uses the exact values, not the rounded 1.96 and 2.576, so it agrees with statistical software to every digit shown.

A quick rule for polls: p(1 − p) is largest at p = 50%, so at 95% confidence the margin can never be more than about 98 ÷ √n percentage points. With 1,000 respondents that is ±3.1.

1. Choose the estimate type: a proportion (a survey percentage), a mean with a known population standard deviation (z), a mean with the standard deviation of the sample (t, the usual case), or the number of respondents needed for a target margin.
2. Enter the values. A proportion is a percentage: type 52 for 52%. For a mean, the sample mean is optional and only adds the interval around it.
3. Choose the confidence level, from 80% to 99.9%. Add the population size only if the sample is a sizeable share of a small, known population.
4. Press Calculate Margin of Error. The working under the results shows every step, and Copy link to this calculation keeps your inputs in a link you can share.

Of 1,000 respondents, 520 support a measure, so the sample proportion is 52%.

1. Standard error = √(0.52 × 0.48 / 1000) = 0.015799.
2. At 95% confidence z = 1.9600 (exactly 1.959964), so the margin of error is 1.959964 × 0.015799 = 0.030965, that is ±3.0965 percentage points, usually quoted as ±3.1.
3. The interval runs from 52% − 3.0965% to 52% + 3.0965%: 48.9035% to 55.0965%.

A sample of 36 measurements has mean 80 and standard deviation s = 12. The standard deviation comes from the sample, so the multiplier is t.

1. Standard error = 12 / √36 = 2.
2. With n − 1 = 35 degrees of freedom the 95% multiplier is t = 2.0301, so the margin of error is 2.0301 × 2 = ±4.0602.
3. The interval is 75.9398 to 84.0602.
4. Using z = 1.96 as if σ were known would give ±3.9199, a margin that is too narrow by about 3.5%. If the 36 measurements had been drawn without replacement from a population of only 200 units, the standard error would shrink by √((200 − 36)/199) and the margin would be ±3.6859.

For a proportion of 50%, the case with the widest margin, the margin at 95% confidence falls only with the square root of the sample size:

Halving the margin takes four times the respondents: 1,000 respondents give ±3.10 and 4,000 give ±1.55. That is why polls stop near 1,000 to 1,500 interviews: the last few hundred respondents buy very little. The 1,067 and 2,401 rows are the sizes that give ±3 and ±2 points. Strictly, ±3 needs 1,067.07 respondents, so the sample size calculator rounds up to 1,068.

Confidence has a price as well. The same poll, 1,000 respondents at 50%, at other confidence levels:

The margin of error quoted for a poll belongs to a single percentage. A lead is the difference between two percentages from the same question, and it is less certain: for two shares p₁ and p₂ in one random sample the standard error of the difference is √((p₁ + p₂ − (p₁ − p₂)²) / n). When the two answers make up nearly all responses and are close, the margin of the lead is almost twice the quoted margin.

Example: 48% against 44% among 1,000 respondents. Each share has a margin of ±3.10, but the standard error of the 4-point lead is √((0.48 + 0.44 − 0.04²) / 1000) = 0.030305, so its margin at 95% confidence is ±5.94 points. A lead of 4 points is well inside that: the poll is compatible with a tie. For two separate polls the variances add instead, √(p₁(1 − p₁)/n₁ + p₂(1 − p₂)/n₂).

For a proportion, or for a mean whose σ is known, the multiplier is z. For a mean with the standard deviation estimated from the same sample it is t with n − 1 degrees of freedom, which approaches z as n grows:

At n = 30 the t multiplier is about 4% larger than z and at n = 10 about 15% larger, so a margin computed with z would be too optimistic. The t table lists the multipliers for other sample sizes and confidence levels.

• Bias. Coverage gaps, non-response, question wording and respondents who answer untruthfully move the estimate in a way that a bigger sample does not fix.
• Weighting and clustering. The formulas assume a simple random sample. The American Association for Public Opinion Research (AAPOR) points out that in most surveys of people or households the classical margin is too small, and that it can be multiplied by the square root of the design effect. With a design effect of 1.5 the ±3.10 of a 1,000-person poll becomes ±3.80.
• Non-random samples. The classical margin of sampling error is defined for probability samples. AAPOR cautions that it may be misleading to report one for an opt-in sample, where any estimate of the likely error rests on a statistical model.
• Subgroups. The margin belongs to the group that was measured: among 250 of the 1,000 respondents it is ±6.20 points, not ±3.10.
• Proportions near 0% or 100%. When fewer than 10 successes or failures are expected, or the interval would leave the range from 0% to 100%, use the Wilson interval of the proportion confidence interval calculator.

In Excel, CONFIDENCE.NORM(alpha, standard_dev, size) returns z × σ / √n, and CONFIDENCE.T(alpha, standard_dev, size) returns t × s / √n, with alpha = 1 minus the confidence level. For the 36 measurements =CONFIDENCE.NORM(0.05,12,36) gives 3.9199, the z margin. Both return the margin of error, not the interval: add and subtract it from the mean. See also the guides to the confidence interval in Excel and to the TI-84 statistics functions.