Last updated: 2026-06-23

What This Calculator Does

This calculator finds the margin of error for estimating a population mean from a sample, along with the confidence interval that surrounds the sample mean. Enter the confidence level, the standard deviation, the sample size, and the observed sample mean.

What a margin of error is

When a poll, survey, or experiment estimates an average from a sample, the estimate is never exact — a different random sample would give a slightly different mean. The margin of error quantifies that sampling uncertainty. It is the "plus or minus" reported alongside results such as "average satisfaction was 7.2 out of 10, ±0.3." Formally, it is the half-width of a confidence interval: the distance from the sample mean to either end of the interval.

The formula

For a population mean with a known (or well-estimated) standard deviation, the margin of error is:

• E — the margin of error (the half-width of the interval)
• z — the critical value from the standard normal distribution, set by the confidence level (1.645 for 90%, 1.96 for 95%, 2.576 for 99%)
• σ — the standard deviation of the data
• n — the sample size

The term $\sigma/\sqrt{n}$ is the standard error of the mean: the standard deviation of the sampling distribution of the sample mean. Multiplying it by the critical value scales it up to the desired confidence level.

The confidence interval is then the sample mean plus or minus this margin:

So a sample mean $\bar{x}$ with margin of error $E$ yields the interval from $\bar{x} - E$ to $\bar{x} + E$. At 95% confidence, this means that if the study were repeated many times, about 95% of the intervals constructed this way would contain the true population mean.

Why the margin shrinks with √n

Because the sample size enters the formula under a square root, the margin of error falls in proportion to $1/\sqrt{n}$ rather than in inverse proportion to $n$ itself. This produces diminishing returns:

To halve the margin of error you must quadruple the sample. This square-root law is why large national surveys still report margins of a few percentage points, and why each additional unit of precision becomes progressively more expensive to buy.

Worked example

Suppose a sample of size 100 has a standard deviation of 15 and a mean of 50, and a 95% confidence interval is wanted. The critical value is 1.96:

The margin of error is ±2.94, and the 95% confidence interval runs from 47.06 to 52.94. You can read this as: based on this sample, the population mean is plausibly anywhere in that range.

When σ is unknown or n is small

The formula above assumes the population standard deviation σ is known, or that the sample is large enough that the sample standard deviation s is a dependable substitute. A common rule of thumb is $n \geq 30$.

When σ is unknown and the sample is small, replace the normal critical value with one from the t-distribution, which has heavier tails to account for the extra uncertainty in estimating σ from the data:

The t-value depends on the degrees of freedom, one fewer than the sample size, and is always slightly larger than the corresponding z-value, so it widens the interval. As n grows, the t-value converges back to z.

Margin of error for a proportion

This calculator covers the margin of error for a mean. For a proportion — for example, the share of respondents who answered "yes" — the standard error is different, and the margin of error becomes:

where $p$ is the sample proportion. This expression is largest when $p$ equals one-half, which is why pollsters often report the conservative "worst case" margin computed at that value.

Related calculators

The Confidence Interval Calculator builds the full interval around your estimate and lets you work with both means and proportions. If you are planning a study and want to choose n in advance to hit a target margin, use the Sample Size Calculator for a Mean to invert the formula and solve for the sample size you need.