Summary: This margin of error calculator computes mean and proportion error with live steps, formulas and a chart. It accepts labeled numeric inputs, works offline through file:// and includes source-backed explanations for students, analysts and researchers.
Margin of error calculator for means and proportions with confidence level, z critical value, formula and interpretation. The calculator works offline, updates instantly and includes a worked example, plain-text formula, MathML, references and structured data.
Default values are loaded. Click any field and edit it; results and chart update automatically.
| Input or setting | Result or interpretation | Use this when |
|---|---|---|
| n=400, p=.52, 95% | E about 0.049 | uncertainty width |
| larger n | smaller margin | uncertainty width |
| higher confidence | larger margin | uncertainty width |
mean: E = z* sigma / sqrt(n) or t* s / sqrt(n); proportion: E = z*sqrt(phat(1 - phat)/n)Margin of error is the half-width of a confidence interval.
n=400, phat=0.52, 95%
E about 0.049, or +/-4.9 percentage points
This page uses the shared statistics core for distribution functions, quantiles and exact integer counting where needed. The formula is shown in plain text so screen readers and search engines can parse it reliably.
Use Load example in the calculator to reproduce this reference result.
{
"tool": "Margin Of Error Calculator",
"input": "n=400, phat=0.52, 95%",
"output": "E about 0.049, or +/-4.9 percentage points",
"formula": "mean: E = z* sigma / sqrt(n) or t* s / sqrt(n); proportion: E = z*sqrt(phat(1 - phat)/n)"
}| Calculator | Example input | Expected output |
|---|---|---|
| Margin Of Error Calculator | n=400, phat=0.52, 95% | E about 0.049, or +/-4.9 percentage points |
What does p = 0.03 mean? If the null hypothesis and model assumptions were true, a result at least this extreme would occur about 3% of the time. The American Statistical Association cautions that a p-value alone does not measure effect size, practical importance or the probability that Hâ‚€ is true.3
For most classroom and professional reports, pair the calculator result with the question you are answering. A mean or median summarizes location, but spread explains consistency. A confidence interval estimates plausible values, while a hypothesis test evaluates compatibility with a null model. Regression and correlation describe association, so they should be reported with a chart and residual or outlier review. When a result is statistically significant, still ask whether the effect is large enough to matter in the real setting.
| Statistic | Small | Medium | Large | Use |
|---|---|---|---|---|
| Cohen's d | 0.2 | 0.5 | 0.8 | t-test effect size |
| Cramér's V | 0.1 | 0.3 | 0.5 | chi-square association |
| |r| | 0.10 | 0.30 | 0.50 | correlation strength |
| R² | 0.01 | 0.09 | 0.25 | variance explained |
The general formula is MOE = critical value × standard error. For a survey proportion at 95% confidence: MOE = 1.96 × √[p(1 − p)/n]. With p = 0.5 and n = 400, that's 1.96 × √(0.25/400) = 1.96 × 0.025 = 0.049, or about 4.9%. For a mean, replace the proportion's SE with σ/√n (or s/√n for a t-based version). The critical value depends on your confidence level — 1.645 for 90%, 1.96 for 95%, 2.576 for 99%.
For a 5% MOE at 95% confidence, assuming the worst-case proportion p = 0.5, you'd use n = z²p(1 − p)/E² = 1.96² × 0.25 / 0.05² = 3.8416 × 0.25 / 0.0025 ≈ 384.16, which rounds up to 385. That's the famous "385 rule" for survey design. If you have a known proportion that's far from 0.5, you can get away with a smaller sample. And if the population is small (under say 10,000), apply the finite population correction to reduce n further.
Higher confidence widens the margin of error. The critical value scales with confidence: z = 1.645 at 90%, 1.96 at 95%, 2.576 at 99%. Since MOE is directly proportional to that z, jumping from 95% to 99% confidence widens the MOE by about 30% for the same sample size. So you're trading precision for certainty. If you want both high confidence and a tight MOE, the only way out is to collect more data — bigger n shrinks the standard error and offsets the larger critical value.
Most of the time, no — surprisingly. Once your population is reasonably large (over a few thousand), MOE depends mainly on sample size, confidence level, and variability, not on the population total. That's why a national poll of 1,000 people can deliver the same precision whether the country has 10 million or 100 million residents. For smaller, finite populations, however, you should apply the finite population correction (FPC), which slightly reduces the MOE. The FPC matters most when your sample is more than 5% of the population.
For most public opinion polls, ±3% to ±5% is considered acceptable at 95% confidence. Tighter margins (±2% or below) are used in close political races where small differences matter. Looser margins (±7% to ±10%) might suit exploratory studies or pilot research with limited budgets. The "right" MOE balances precision against cost: every halving of MOE roughly quadruples the sample size you need. Match the MOE to your decision — if a 3-point swing changes your action, your MOE needs to be smaller than 3 points.
Three reliable levers. First, increase your sample size — MOE shrinks proportionally to 1/√n, so quadrupling n halves the MOE. Second, lower your confidence level (95% → 90% gives a smaller critical value, but you accept more risk of being wrong). Third, reduce variability if you can — through better measurement tools, stratified sampling, or more homogeneous subgroups. One thing MOE does not fix is bias: collecting bigger non-random samples just makes biased estimates more confidently wrong, so good sampling design matters as much as sample size.
Use MOE = z × √[p(1 − p)/n]. Pick the z critical value for your confidence level (1.96 for 95%), plug in your sample proportion p, and divide by sample size n. If you don't yet have data and want a conservative estimate, set p = 0.5 — that maximises p(1 − p) = 0.25 and gives the largest possible MOE for that sample size. Example: at 95% confidence with n = 500 and p = 0.5, MOE = 1.96 × √(0.25/500) ≈ 0.0438, or roughly 4.4%.
At 95% confidence with the worst-case proportion p = 0.5, the MOE is 1.96 × √(0.25/1000) ≈ 0.031, or about ±3.1%. That's why news outlets reporting national polls of 1,000 respondents typically cite a "±3 percentage point" margin. The exact MOE shifts a bit if the actual proportion differs from 0.5 — for p = 0.2 or 0.8, the MOE drops to roughly ±2.5%. Also remember this calculation ignores non-response bias and design effects, which can be larger than the stated MOE in real surveys.