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Margin of Error Calculator

Free margin of error calculator. Proportion and mean MOE, required sample size, confidence interval,

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Why This Statistical Analysis Matters

Why: Statistical calculator for analysis.

How: Enter inputs and compute results.

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STATISTICSSurveys & Polls

Margin of Error — Surveys & Polls

Calculate MOE, required sample size, and confidence intervals. Proportion and mean. Simple or advanced mode with FPC.

Confidence Interval →Sample Size →

Real-World Scenarios — Click to Load

Simple: p + n + CL → MOE

moe_results.sh
CALCULATED
$ margin_of_error --mode="moe" --type="proportion" --cl=0.95
MOE
±3.10%
95% CI
[46.90%, 53.10%]
z*
1.960
Share:
Margin of Error Result
MOE = ±3.10%
95% CI: [46.90%, 53.10%]
numbervibe.com/calculators/statistics/margin-of-error-calculator

Calculation Breakdown

COMPUTATION
Standard Error
0.0158
SE = √(p̂(1−p̂)/n) = √(0.5×0.5000/1000)
DECISION
z* (critical value)
1.9600
95% confidence
MOE
±3.10%
z* × SE
CONFIDENCE INTERVAL
95% CI
[46.90%, 53.10%]

MOE vs Sample Size

MOE vs Confidence Level

Confidence Interval

For educational and informational purposes only. Verify with a qualified professional.

Key Takeaways

  • • MOE for proportion: MOE = z × √(p̂(1−p̂)/n). If p̂ unknown, use p̂ = 0.5 (maximum MOE).
  • • MOE for mean: MOE = z × σ/√n (σ known) or t × s/√n (σ unknown, df = n−1).
  • • Finite population correction: MOE_fpc = MOE × √((N−n)/(N−1)). Use when sampling > 5% of population.
  • • Required sample size (proportion): n = (z² × p̂(1−p̂)) / MOE². With FPC: n = n₀N / (n₀ + N − 1).
  • • Required sample size (mean): n = (z × σ / MOE)².
  • • z-values: 90% ≈ 1.645, 95% ≈ 1.96, 99% ≈ 2.576.

Practical Polling Interpretation

🗳️Political polls often report ±3% at 95% confidence. That means 19 out of 20 such polls would capture the true proportion.
📊Using p̂ = 0.5 gives the most conservative (largest) MOE for proportions — safe when you have no prior estimate.
📉MOE decreases with √n. Doubling sample size only reduces MOE by about 29%.
🏥Clinical trials use MOE for mean outcomes (e.g., blood pressure change) with t-distribution when σ is unknown.
📧Email open rates around 20–30% need ~1000+ sends for ±3% MOE at 95% confidence.
🎓For small populations (e.g., class of 200), finite population correction reduces required sample size.

Formulas

MOE_proportion = z × √(p̂(1−p̂)/n)

Standard error of proportion

MOE_mean = z × σ/√n (σ known) or t × s/√n (σ unknown)

Standard error of mean

FPC = √((N−n)/(N−1))

Finite population correction

n = (z² × p̂(1−p̂)) / MOE²

Required sample size for proportion

Frequently Asked Questions

What confidence level should I use?

95% is standard for most surveys. Use 99% when you need higher certainty; 90% when a rough estimate suffices.

Why use p̂ = 0.5 when proportion is unknown?

p̂(1−p̂) is maximized at 0.5, giving the largest (most conservative) MOE. Safe for planning.

When do I need finite population correction?

When your sample is more than 5% of the population. Common in small populations (schools, organizations).

What is the difference between MOE and standard error?

MOE = critical value × SE. MOE is the half-width of the confidence interval.

Why does doubling sample size not halve MOE?

MOE ∝ 1/√n. To halve MOE, you need 4× the sample size.

Quick Reference: Common MOE Values

At 95% confidence, p̂=0.5: n=400 → ±4.9%, n=600 → ±4.0%, n=1000 → ±3.1%, n=1500 → ±2.5%, n=2400 → ±2.0%. Use these as benchmarks when planning surveys.

Disclaimer: MOE reflects sampling error only. Non-response, measurement error, and selection bias can affect survey accuracy beyond what MOE captures.

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