Margin of Error Calculator

About the Margin of Error Calculator

The margin of error (MOE) quantifies the uncertainty in a survey or poll result. When a poll reports "52% ± 3%," the 3% is the margin of error, meaning the true population value is likely between 49% and 55%. Understanding and calculating MOE is essential for interpreting any survey, poll, or sample-based estimate.

This calculator works in two modes: (1) calculate the margin of error from a given sample, or (2) determine the sample size needed to achieve a desired margin of error. It handles both proportion estimates (surveys, polls) and mean estimates (continuous measurements), with optional finite population correction for sampling from known-size populations.

The calculator also shows how MOE varies with confidence level and sample size, making it easy to explore trade-offs between precision, confidence, and data collection cost. Use the sample-size view to see how much larger your survey needs to be to narrow the interval, and use finite population correction when sampling from a relatively small known population.

When This Page Helps

Every sample-based estimate has uncertainty. The margin of error makes that uncertainty concrete and interpretable. This calculator handles both directions — from sample to MOE, and from desired MOE to required sample size — saving time in study planning. The interactive tables let you see how changing confidence level or sample size affects precision.

How to Use the Inputs

1. Choose the mode: calculate MOE from sample or find required sample size.
2. Enter the confidence level (commonly 0.95 for 95%).
3. For MOE calculation, enter sample size and proportion (or mean/SD).
4. For sample size calculation, enter desired MOE and expected proportion.
5. Optionally enter population size for finite population correction.
6. Review the margin of error and confidence interval.
7. Use the comparison tables to explore different scenarios.

Example Calculation

Tips & Best Practices

• Use p = 0.5 when the expected proportion is unknown — this gives the most conservative (largest) sample size and MOE.
• The margin of error decreases with the square root of n: to halve the MOE, you need 4× the sample size.
• Finite population correction matters when sampling more than ~5% of the population.
• MOE only covers random sampling error — not response bias, non-response bias, or measurement error.
• Higher confidence levels require larger margins of error (or larger samples). 99% CI is wider than 95% CI.
• For proportions near 0 or 1, the normal approximation may be poor. Use Wilson or Clopper-Pearson intervals instead.

Understanding the ±3% in News Polls

When media reports a poll with "margin of error ±3%," this means at the stated confidence level (usually 95%), the true population proportion is expected to fall within 3 percentage points of the reported figure. If Candidate A polls at 48% ± 3%, the true support is likely between 45% and 51%. If the race is within the margin, it's a statistical tie.

The Sample Size Sweet Spot

There's a diminishing returns relationship between sample size and precision. Going from n=100 to n=400 cuts MOE in half. But going from n=1,000 to n=4,000 also only cuts MOE in half. Most surveys find n=1,000-1,500 a practical sweet spot, yielding ±3% at 95% confidence. Beyond that, additional precision comes at steep cost.

MOE for Subgroup Analysis

The margin of error increases for subgroups within a sample. If you survey 1,000 people but analyze only the 200 who are age 18-24, the MOE for that subgroup is much larger (based on n=200, not n=1,000). Always check whether subgroup analyses have adequate sample sizes.

Frequently Asked Questions

• What is margin of error?

  It's the maximum expected difference between the sample statistic and the true population parameter at a given confidence level. A MOE of ±3% means the true value is likely within 3 percentage points of the sample estimate.

• Why does sample size matter more than population size?

  For large populations, MOE depends almost entirely on sample size, not population size. Polling 1,000 people gives the same accuracy whether the population is 1 million or 100 million. FPC only matters for small populations.

• What confidence level should I use?

  95% is standard and widely expected. Use 99% for high-stakes decisions (medical, legal). Use 90% for exploratory research where slightly more risk is acceptable. Higher confidence = wider MOE.

• How do I reduce the margin of error?

  Three ways: increase sample size (most common), lower the confidence level (trade-off), or reduce variability in the population (often not controllable). Doubling the sample roughly reduces MOE by 29%.

• What is the finite population correction?

  When you sample a substantial fraction of the total population, the standard MOE formula overestimates uncertainty. The FPC factor √((N−n)/(N−1)) reduces the MOE. It's negligible when n < 0.05×N.

• Does margin of error account for all sources of error?

  No. MOE only captures random sampling error — the variability from taking a random sample. It does not account for non-response bias, question wording effects, interviewer bias, or coverage error, which can be much larger.