Margin of Error Calculator and Formula Explained

This margin of error calculator estimates the maximum sampling error for a survey proportion from the sample size and confidence level you choose. Enter the number of responses, the confidence level, and an estimated proportion; it returns the margin of error in percentage points.

Formula

For a proportion measured in a simple random sample, the approximate margin of error is:

MOE = z × √[p(1 − p) / n]

• MOE is the margin of error as a proportion; multiply it by 100 to express it in percentage points.
• z is the critical value for the selected confidence level: 1.645 for 90%, 1.96 for 95%, or 2.576 for 99%.
• p is the estimated proportion expressed as a decimal, from 0 to 1.
• n is the number of eligible responses included in the sample.

The calculator accepts p as a percent, so its first output directly gives percentage points. The approximate interval is p ± MOE. This is the standard normal-approximation calculation for a proportion, not a universal margin-of-error formula for every kind of statistic.

Worked Example

Using the defaults—n = 1,000, a 95% confidence level (z = 1.96), and p = 50%—the calculation is:

1. Convert the estimated proportion to a decimal: p = 50 / 100 = 0.50.
2. Calculate its standard error: √[0.50 × (1 − 0.50) / 1,000] = √0.00025 ≈ 0.015811.
3. Multiply by the critical value: 1.96 × 0.015811 ≈ 0.03099.
4. Convert to percentage points and round to two decimals: MOE = 3.10 percentage points.

The approximate bounds are 50% − 3.10% = 46.90% and 50% + 3.10% = 53.10%. Thus the calculator reports a margin of error of 3.10 percentage points, with bounds of 46.90% to 53.10%.

What the Margin of Error Means

A margin of error describes sampling uncertainty: the difference that could reasonably arise because a survey asked a sample rather than every member of the population. With a 95% confidence method, the interpretation is about the long-run performance of the method: across repeated comparable samples, about 95% of intervals constructed this way would contain the true population proportion, assuming the sampling model and its conditions hold.

It does not mean there is a 95% probability that this particular fixed interval contains the true value. Nor does it guarantee that a poll is accurate within the stated amount. Poor question wording, nonresponse, coverage gaps, weighting choices, and other sources of bias can shift results in ways this sampling margin does not capture.

Choosing the Inputs

• Sample size: Use the number of completed, usable responses relevant to the question—not the number invited or the total population. If different questions have different response counts, calculate each using its own n.
• Confidence level: A higher confidence level uses a larger z value and produces a wider margin. The common 90%, 95%, and 99% choices correspond to the values shown in the calculator.
• Estimated proportion: Enter the expected share as a percent between 0 and 100. If you do not know it in advance, 50% is a conservative planning choice for this formula: p(1 − p) is largest at 0.50, giving the widest margin for a fixed sample size and confidence level.

For planning a survey when the population proportion is unknown, use p = 50% to avoid understating the margin due to an optimistic guess. For interpreting results, use the observed percentage for the specific response option being reported.

What Changes the Result

At a fixed confidence level and proportion, a larger sample narrows the margin, but not in direct proportion to the sample size. Because the formula divides by the square root of n, cutting the margin roughly in half requires about four times as many responses. Raising the confidence level widens the interval. The margin is widest near 50% and narrower near 0% or 100% under this approximation.

This formula assumes observations are independent and the sample behaves like a simple random sample. Clustered, stratified, or heavily weighted surveys may need a design effect or a survey-specific standard error. If a sampling fraction is substantial because a large share of a small finite population was surveyed, a finite population correction may also be relevant. Follow the survey designer’s method when available rather than treating this basic calculation as a substitute.

Common Mistakes and Limits

• Confusing percentage with percentage points: A result of 3.10 percentage points around 50% gives 46.90%–53.10%. It is not a relative change of 3.10% of 50.
• Applying it to every average: This formula is for a proportion or yes/no outcome. A mean, such as average height or spending, uses a standard error based on the data’s standard deviation and an appropriate critical value.
• Ignoring sampling bias: A very large convenience sample can still be unrepresentative. Increasing n reduces random sampling error, not systematic bias.
• Trusting the normal approximation at extremes: When the sample is small or the estimated proportion is close to 0% or 100%, the simple symmetric interval can perform poorly and may even imply impossible bounds. Consider a Wilson or exact binomial interval instead.
• Rounding too early: Keep full precision through the calculation and round only the final result. The displayed bounds are approximate and may not add exactly because of rounding.

Frequently Asked Questions

How do I calculate margin of error by hand?
For a proportion, convert the percentage to a decimal, calculate √[p(1 − p) / n], multiply by the confidence-level z value, then multiply by 100 to express the answer in percentage points.

What is the margin of error at 95% confidence?
It depends on the sample size and estimated proportion. For example, with 1,000 responses and an estimated proportion of 50%, the approximate 95% margin is 3.10 percentage points.

Does a bigger sample always give a smaller margin of error?
Under the same sampling assumptions, yes. The margin falls in proportion to 1/√n, so increasing the sample fourfold approximately halves it.

Can the margin of error be zero?
It approaches zero as the sample size grows, but for a finite sample with a proportion between 0 and 1, this formula gives a positive value. A reported zero may simply reflect rounding or a proportion at an endpoint.

Does margin of error include survey bias?
No. It estimates random sampling uncertainty under the stated assumptions. It does not account for bias from nonresponse, an unrepresentative sampling frame, question wording, or other systematic problems.

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in n | Sample size | people | 1000 | 1
sel c | Confidence level | 90%=1.645, 95%=1.96, 99%=2.576
in p | Estimated proportion | % | 50 | 0
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