Margin of Error Calculator

The margin of error $E$ is the half-width of a confidence interval — how far a sample estimate can plausibly sit from the true population parameter.

Mean, population $\sigma$ known:

$E = z^* \cdot \frac{\sigma}{\sqrt{n}}$

Mean, $\sigma$ unknown (the usual case):

$E = t^*_{n-1} \cdot \frac{s}{\sqrt{n}}$

Proportion:

$E = z^* \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$

Symbols: $n$ is the sample size, $\sigma$ the population standard deviation, $s$ the sample standard deviation, $\hat{p}$ the sample proportion, and $z^*$ or $t^*$ the critical value for the chosen confidence level.

The interval is then $\text{estimate} \pm E$. Because $E$ shrinks like $1/\sqrt{n}$, quadrupling the sample only halves the margin of error.

Critical values

Confidence	$z^*$
90%	$1.645$
95%	$1.96$
99%	$2.576$

Use $z^*$ when $\sigma$ is known; use $t^*$ with $df = n - 1$ whenever you estimate the spread from the sample with $s$. At 99% confidence with $n = 25$, $t^*_{24} = 2.797$ — noticeably wider than $z^* = 2.576$, which is exactly the penalty for not knowing $\sigma$.

Standard error vs margin of error

The standard error is $\sigma/\sqrt{n}$ (or $\sqrt{\hat{p}(1-\hat{p})/n}$). The margin of error is that standard error multiplied by the critical value. They are different numbers, and only the second one carries a confidence level.

Sample size for a target margin

Solve for $n$:

$n = \left(\frac{z^*\sigma}{E}\right)^2 \qquad\qquad n = \hat{p}(1-\hat{p})\left(\frac{z^*}{E}\right)^2$

With no prior estimate of $\hat{p}$, use $0.5$ — it maximises $\hat{p}(1-\hat{p})$ and so is the safe choice. Always round $n$ up.

These formulas are valid only when:

• Random sampling: the data come from a simple random sample or a randomised experiment. Margin of error quantifies sampling variability only — it says nothing about bias from non-response, bad question wording, or a self-selected panel.
• Independence: observations are independent. When sampling without replacement, keep $n \le 10\%$ of the population.
• Approximate normality: for a mean, either the population is roughly normal or $n \ge 30$ so the Central Limit Theorem applies. For a proportion, check $n\hat{p} \ge 10$ and $n(1-\hat{p}) \ge 10$.

Frequent errors

• Using $z^*$ when $\sigma$ is unknown and $n$ is small — that understates $E$; use $t^*$.
• Dividing by $n$ instead of $\sqrt{n}$.
• Grabbing the one-tailed critical value: a two-sided 95% interval uses $1.96$, not $1.645$.
• Assuming a larger population demands a larger sample. Population size is almost irrelevant; $n$ is what drives $E$.