Margin of Error Calculators

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The margin of error (MOE) quantifies the uncertainty in a sample estimate — how far the sample statistic (e.g., poll percentage) might differ from the true population value. It defines the half-width of a confidence interval: CI = estimate ± MOE. For a proportion, MOE = z × √(p(1−p)/n), where z is the critical value for the desired confidence level (1.96 for 95%), p is the estimated proportion, and n is sample size. MOE decreases as sample size increases and increases as confidence level increases. It is widely used in political polling, quality control, medical surveys, and market research.

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Margin of Error Formula

MOE = z × √(p(1-p)/n)

z = 1.645 (90% confidence); 1.960 (95% confidence); 2.576 (99% confidence). p = estimated proportion (use 0.5 for maximum MOE when p unknown). n = sample size. Example: n = 1000, p = 0.5, 95% confidence: MOE = 1.96 × √(0.25/1000) = 1.96 × 0.01581 = 0.031 = ±3.1%.

Confidence Interval

CI = p̂ ± MOE

Example: poll shows 52% support with MOE ±3%: CI = (49%, 55%). This means we are 95% confident the true population proportion is between 49% and 55%.

Factors Affecting MOE

  • Sample size: MOE ∝ 1/√n — quadrupling n halves MOE
  • Confidence level: Higher confidence (99% vs. 95%) requires larger z → larger MOE for same n
  • Population proportion: MOE is maximum when p = 0.5; smaller when p is near 0 or 1

Sample Size for Target MOE

n = z² × p(1-p) / MOE²

For MOE ≤ ±3% (0.03) at 95% confidence: n = 1.96² × 0.25 / 0.03² = 3.84 × 0.25 / 0.0009 = 1067. This is why many national polls use n ≈ 1000.

Glossary

Margin of Error (MOE)
The half-width of a confidence interval: MOE = z × √(p(1−p)/n); quantifies uncertainty in a sample estimate; ±3% for a typical poll of n ≈ 1000 at 95% confidence.
Confidence Interval (CI)
A range (estimate ± MOE) that captures the true population parameter with a specified probability (e.g., 95%); interpreted as: 95% of such intervals from repeated sampling would contain the true value.
Critical Value (z)
The z-score corresponding to the desired confidence level: z = 1.645 (90%); 1.960 (95%); 2.576 (99%); multiplied by the standard error to compute MOE.

Frequently Asked Questions

Margin of error (MOE) is the half-width of a confidence interval — the maximum expected difference between a sample statistic and the true population value. Formula: MOE = z × √(p̂(1−p̂)/n), where z is the critical value (1.96 for 95% CI), p̂ is the sample proportion, and n is sample size. Example: 600 survey respondents, 45% answered yes: MOE = 1.96 × √(0.45×0.55/600) = 1.96 × 0.0203 = 0.0398 ≈ ±4.0%. Report: 45% ± 4%.

MOE ∝ 1/√n — doubling sample size reduces MOE by a factor of √2 (about 29%); quadrupling n halves MOE. Formula: MOE = z × √(p(1−p)/n). To cut MOE from ±4% to ±2%, you need 4× the sample size. This is why polls with very small samples (n = 100) have large MOE (±10%), while national polls with n = 1000 achieve MOE ≈ ±3%, and exit polls with n = 10,000 achieve MOE ≈ ±1%. Beyond n ≈ 1500, additional sample gives diminishing returns on MOE reduction.

Confidence interval (CI) = estimate ± margin of error. MOE is the half-width. A 95% CI means: if you repeated the survey 100 times with different random samples, approximately 95 of the 100 confidence intervals would contain the true population value. Example: poll result 48% ± 3% at 95% CI means the CI is (45%, 51%). It does NOT mean there is a 95% probability the true value is between 45% and 51% — the true value is fixed; it either is or isn't in that interval. The 95% refers to the long-run reliability of the method.

Required sample size for a desired MOE: n = z² × p(1−p) / MOE². For 95% confidence (z=1.96), worst case p=0.5, MOE=0.03: n = 1.96² × 0.25 / 0.03² = 3.8416 × 0.25 / 0.0009 = 0.9604/0.0009 ≈ 1068 people. For ±2% MOE: n = 3.8416 × 0.25 / 0.0004 = 2401 people. For ±1% MOE: n ≈ 9604. Note: this formula assumes simple random sampling. Non-probability sampling, cluster designs, and non-response can inflate the effective MOE beyond these theoretical values.