Margin of Error: Formula and Calculation

A margin-of-error calculator estimates how far a survey result may be from the corresponding population value because of sampling variability. Enter the sample size, confidence level, and (for a proportion) the estimated percentage; the result is the margin of error in percentage points.

Formula

For a sample proportion, the usual approximate margin of error is:

MOE = z × √[p(1 − p) / n]

• MOE is the margin of error, expressed as a proportion; multiply by 100 to report percentage points.
• z is the critical value for the selected confidence level: approximately 1.645 for 90%, 1.96 for 95%, and 2.576 for 99%.
• p is the estimated proportion, entered here as a percentage and converted to a decimal in the calculation.
• n is the number of independent observations in the sample, a count with no unit.

The calculator uses the normal-approximation formula for a proportion and reports the margin in percentage points. Its bounds are the estimate minus and plus that margin. The estimate should normally be between 0% and 100%; the approximate interval can extend beyond those limits near an extreme.

Worked Example

With the calculator’s defaults, the sample contains 1,000 people, the estimated proportion is 50%, and the confidence level is 95%. Thus p = 0.50 and z = 1.96.

1. Calculate the standard error: √[0.50 × (1 − 0.50) / 1,000] = √0.00025 ≈ 0.015811.
2. Multiply by the confidence multiplier: 1.96 × 0.015811 ≈ 0.03099.
3. Convert to percentage points: 0.03099 × 100 ≈ 3.10.

The margin of error is 3.10 percentage points. The corresponding approximate bounds are 50% − 3.10% = 46.90% and 50% + 3.10% = 53.10%.

What the Result Means

A margin of error describes sampling uncertainty under the assumptions of the method; it is not a guarantee that the true population value lies inside the calculated bounds. A 95% confidence level refers to the long-run performance of the procedure: across repeated comparable samples, about 95% of intervals constructed this way would cover the population proportion, assuming the sampling model is appropriate.

If a poll reports 50% with a margin of error of 3 percentage points, its conventional interval is about 47% to 53%. The margin is stated in percentage points, not as a relative percentage change. For example, 50% plus or minus 3 percentage points means 47% to 53%, not 48.5% to 51.5%.

How Sample Size and Confidence Affect It

The standard error contains √n in the denominator, so larger samples generally produce smaller margins, but with diminishing returns. To cut the margin in half while keeping the confidence level and estimated proportion unchanged, the sample size must be about four times as large. Raising the confidence level increases z and therefore widens the margin.

The value of p also matters. For a proportion, p(1 − p) is largest at p = 0.50, so using 50% gives the most conservative margin when the proportion is unknown. If no reliable estimate is available, this is a common planning assumption. It does not mean that 50% is the expected result; it is a way to avoid understating the maximum sampling margin for a given sample size.

When This Calculation Applies—and Its Limits

This formula is intended for a proportion estimated from a reasonably large, independent random sample, such as the share of respondents choosing an option. The normal approximation is most dependable when the sample contains enough observations in both outcome categories. With small samples or proportions very close to 0% or 100%, specialized methods, such as Wilson or exact binomial intervals, may be more suitable.

A conventional margin of error covers only the sampling variability represented by the formula. It does not automatically account for nonresponse, biased question wording, inaccurate answers, coverage gaps, data-entry errors, or other survey design problems. A large sample can still produce a misleading estimate if the people sampled do not represent the population of interest.

Complex designs also need care. Clustering, unequal selection probabilities, and weighting can increase or otherwise alter uncertainty. A simple-random-sample calculation based only on the number of respondents may then be too optimistic; analysts generally need a design-aware variance estimate or an effective sample size.

Common Calculation Mistakes

• Entering a count instead of a percentage: The calculator expects p as a percentage from 0 to 100, such as 42 for 42%.
• Confusing percent with percentage points: A margin of 3 points around 42% gives 39% to 45%.
• Using the wrong sample size: Use the number of observations relevant to that specific estimate, not automatically the survey’s total size.
• Rounding too early: Keep full precision in intermediate steps and round the final result to a sensible number of decimal places.
• Calling the margin the total error: It measures a particular source of uncertainty and does not include every possible source of survey error.

Frequently Asked Questions

How do you calculate margin of error by hand?
For a proportion, calculate p(1 − p) / n, take the square root, and multiply by the z-value for the desired confidence level. Multiply by 100 to express the answer in percentage points.

What is the margin of error for a 1,000-person sample?
It depends on the confidence level and estimated proportion. At 95% confidence and p = 50%, the approximate margin is 3.10 percentage points under the simple-random-sample assumptions used here.

Does a larger sample always mean a smaller margin of error?
Under the same sampling model and confidence level, increasing the sample size reduces the calculated sampling margin. It does not necessarily reduce bias or other survey errors.

Can a margin of error be calculated for an average?
Yes, but the proportion formula above is not the right one. For a mean, a common form is a critical t- or z-value multiplied by the standard error of the mean; calculating it requires the sample standard deviation and sample size, as well as the chosen confidence level.

Why do some polls have no stated margin of error?
A poll may use a nonprobability sample, a complex design, or a method for which the conventional simple-random-sample margin is not appropriate. The pollster may instead report a model-based uncertainty measure or explain why a standard margin is unavailable.

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in n | Sample size | people | 1000 | 1
in p | Estimated proportion | % | 50 | 0
sel c | Confidence level | 90%=1.645, 95%=1.96, 99%=2.576
out moe | Margin of error | percentage points | c * 100 * sqrt((p / 100) * (1 - p / 100) / n) | 2
out low | Lower bound | % | p - moe | 2
out high | Upper bound | % | p + moe | 2