Calculators

Margin of Error Calculator

Margin of error for a survey proportion or a sample mean — with confidence level, optional finite-population correction, auto Z/t switch and a shaded normal-curve diagram.

Margin of Error — Proportion (survey / poll)

MoE = z × √(p(1 − p) / n), optionally multiplied by the finite-population correction √((N − n) / (N − 1)).

Margin of Error — Mean

MoE = critical × s / √n. Auto-switches to Student's t for small samples (n < 30), matching the Confidence Interval Calculator.

Method:

Margin of error explained, step by step

Margin of error is the ± range a poll or study reports at a given confidence level. Each card below covers one piece the calculator uses on the proportion and mean tools.

1. What margin of error is — critical × standard error

Margin of error is the ± half-width added to a sample estimate to express sampling uncertainty at a chosen confidence level. It combines two ingredients: the standard error (how noisy the estimate is) and a critical value (how confident you want to be).

MoE = critical × SE
criticalz or t for your confidence level·SEstandard error of the estimate
Example
Given
SE = 0.015
z = 1.96
Substitute
MoE = 1.96 × 0.015
Answer
MoE ≈ ±0.030

2. Proportion formula — √(p(1 − p)/n)

For yes/no survey questions the standard error is √(p(1 − p)/n). When the true p is unknown, use p = 0.5 to get the widest (most conservative) MoE for planning. Add the finite-population correction if the sample is a large fraction of the population.

MoE = zα/2 · √( p(1 − p) / n )
Example
Given
p = 0.60
n = 1000, 95% CI
Substitute
1.96·√(0.24/1000)
Answer
MoE ≈ ±3.04%

3. Mean formula — t for small n, Z for large n

For continuous measurements the standard error is s/√n. Use the Student t critical value with df = n − 1 when the sample is small; once n ≥ 30 the t and Z critical values almost agree and Z is a fine shortcut.

MoE = tα/2, n−1 · ( s / √n )
ssample standard deviation
Example
Given
s = 10
n = 25, 95% CI
Substitute
t₀.₀₂₅, ₂₄ · 10 / √25
Answer
MoE ≈ ±4.13

4. Sample size effect — shrinks with √n

Because n sits under a square root, halving MoE requires quadrupling the sample. Going from n = 400 to n = 1600 only halves the ± width. That's why polls hit a sharp diminishing return around 1,000–2,000 respondents.

MoE ∝ 1 / √n
Example
Given
n = 100 vs 1600
p = 0.5, 95% CI
Substitute
±9.8% vs ±2.45%
Answer
16× n → 4× tighter

Features of this calculator

  • Proportion and mean tools on the same page.
  • Preset confidence levels (80/85/90/95/98/99/99.5/99.9%) plus a custom-level input.
  • Finite-population correction on the proportion tool for surveys of small populations.
  • Auto-switches between t (small n) and Z (n ≥ 30) on the mean tool.
  • Critical-value curve visualising the α/2 tails and a horizontal ± bar for the estimated MoE.
  • Show / hide step-by-step working covering the SE, critical value and multiplication.
+Is margin of error the same as standard error?
No. Standard error is the standard deviation of the sampling distribution (s / √n or √(p(1−p)/n)). Margin of error is the standard error multiplied by a critical value (z or t) chosen for your confidence level — MoE = critical × SE.
+Can margin of error be larger than the estimate itself?
Yes, especially with very small samples or rare events. A poll that finds 3% support with n = 100 has a 95% MoE around ±3.3%, so the interval crosses zero — a signal that the sample is too small to say anything precise.
+How does confidence level change the number?
Higher confidence widens the interval. Going from 95% to 99% confidence (z from 1.960 to 2.576) enlarges the margin of error by about 31% for the same sample size.
+Do I need to know the population size?
Only when the sample is a substantial fraction of a small population (n / N > ~5%). Otherwise the finite-population correction is essentially 1 and can be omitted.
+How is the mean version different from the proportion version?
The proportion version uses √(p(1−p)/n) as the standard error and always uses Z. The mean version uses s / √n and switches to the Student t-distribution when n is small, because s is itself estimated from the data.
+Does margin of error tell me the poll is 'right'?
No — it only bounds sampling variability. It cannot detect bad questionnaire design, non-response bias, or a non-random sample. Those errors need methodology fixes, not a wider interval.
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