Margin of Error Calculator: Formula and Examples

A margin of error calculator estimates how far a survey result may be from the population value because only a sample was measured. Enter the sample size, confidence level, and—when estimating a proportion—the observed percentage; the result is a margin in percentage points. This simple calculator uses the conservative 50% proportion assumption by default.

Formula

For a proportion, the approximate margin of error is:

MOE = z × √[p(1 − p) / n]

• MOE is the margin of error as a proportion; multiply by 100 to express it in percentage points.
• z is the standard-normal critical value for the selected confidence level. Common two-sided values are 1.645 for 90%, 1.96 for 95%, and 2.576 for 99%.
• p is the sample proportion written from 0 to 1, such as 0.50 for 50%.
• n is the number of independent observations included in the estimate.

The calculator uses 1.96, the usual critical value for 95% confidence. The confidence-level field is shown to identify the intended level; if you change it, use the matching z value in the formula. For a sample mean, the analogous formula is MOE = t × s / √n, where s is the sample standard deviation and the t critical value depends on the degrees of freedom. This calculator is specifically for proportions.

Worked Example

With the defaults—sample size 1,000, confidence level 95%, and proportion 50%—the estimate is:

1. Convert 50% to a proportion: p = 0.50.
2. Calculate the standard error: √[0.50 × (1 − 0.50) / 1,000] = √0.00025 ≈ 0.01581.
3. Multiply by 1.96: 1.96 × 0.01581 ≈ 0.03099.
4. Convert to percentage points: 0.03099 × 100 ≈ 3.10 percentage points.

For a measured proportion of 50%, the corresponding approximate interval is 50% − 3.10 to 50% + 3.10, or 46.90% to 53.10%. The calculator rounds displayed results to two decimal places.

What the Margin of Error Means

A margin of error describes sampling uncertainty under a statistical model. If a properly designed survey reports 50% with a margin of error of about 3.1 percentage points at 95% confidence, its confidence interval is approximately 46.9% to 53.1%. “Percentage points” matters: the interval is not 3.1% of 50. A 3.1% relative change from 50% would be a different calculation.

In repeated random samples made under the same conditions, a 95% confidence procedure would produce intervals containing the true population proportion about 95% of the time. It does not mean there is a 95% probability that this particular fixed interval contains the true value. The true population value is fixed; the interval varies from sample to sample.

How Sample Size and Confidence Affect It

For a fixed confidence level and proportion, margin of error falls in proportion to 1/√n. To halve the margin, you generally need about four times as many observations—not twice as many. A higher confidence level uses a larger z value and therefore gives a wider interval. For example, a 99% interval is wider than a 95% interval based on the same data.

The value of p also affects the result. The expression p(1 − p) is largest at p = 0.50, so 50% gives the largest standard error for a given sample size. When the proportion is unknown, using 50% is a conservative planning choice. If the actual estimate is near 0% or 100%, a more suitable interval method may be needed; the simple formula can behave poorly near those boundaries.

When to Use It—and What It Leaves Out

This calculation is useful for a simple random sample when reporting a proportion, such as the share of respondents selecting an answer. It does not by itself show whether a survey is representative or well conducted. A large sample can still be biased if people were selected in a way that systematically excludes part of the population or if many selected people do not respond.

The formula also assumes observations are independent and uses a normal approximation. Survey designs involving clusters, unequal weights, or stratification may require a design effect or specialized software. In a sample drawn without replacement from a small, known population, a finite population correction can reduce the standard error. For a sample mean, use the standard deviation and an appropriate t critical value rather than this proportion formula.

Common Mistakes

• Confusing percent with percentage points: an interval around 50% with a 3-point margin runs from 47% to 53%, not from 48.5% to 51.5%.
• Using the wrong n: count observations that actually contribute to the estimate, not the number invited or the total population. If responses are weighted or grouped, the effective sample size may differ.
• Treating the result as a guarantee: the margin reflects modeled random sampling error, not every source of uncertainty or bias.
• Rounding too early: keep full precision through the calculation and round only the final displayed margin.
• Comparing overlapping intervals as a significance test: overlap alone does not settle whether two estimates differ; a direct test of the difference may be required.

Frequently Asked Questions

How do I calculate margin of error by hand?
For a proportion, compute the standard error √[p(1 − p) / n], multiply it by the z value for your confidence level, and multiply by 100 if you want percentage points.

What is the margin of error for a 95% confidence level?
There is no single value: it depends on sample size and the estimated proportion. For a proportion, use z ≈ 1.96 in the formula.

Does a larger sample always reduce margin of error?
It reduces the sampling margin under the model when other inputs stay fixed. It does not automatically correct selection bias, measurement problems, or nonresponse.

Can I calculate a margin of error without a sample size?
No. Sample size is part of the standard error. For a proportion, you also need an estimated proportion or a planning assumption such as 50%, plus a confidence level.

Is margin of error the same as standard error?
No. Standard error estimates the variability of a statistic across samples. Margin of error is a chosen critical-value multiple of that standard error for a specified confidence level.

Related

• Calculating a Z Test Statistic: Formula and Examples
• Calculating the Median: Formula, Steps, and Examples
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in n | Sample size | people | 1000 | 1
in c | Confidence level | % | 95 | 0
in p | Sample proportion | % | 50 | 0
out moe | Margin of error | percentage points | 100 * (1.96 * sqrt((p / 100) * (1 - p / 100) / n)) | 2
out lower | Lower interval bound | % | p - moe | 2
out upper | Upper interval bound | % | p + moe | 2