Margin of Error Calculator

Calculate the margin of error for a survey or poll. Find how accurate your sample results are based on sample size, confidence level, and proportion. Used in political polling and market research.

📊 Statistics📐 MOE = z* × √(p(1-p)/n)🔢 Math
Sample size n
Sample proportion p (0.5 if unknown)
Confidence level
Please enter valid values.

Formula & Reference

VariableSymbolFormulaUnits
Margin of Error Calculator—MOE = z* × √(p(1-p)/n)± percent

Step-by-Step Examples

Example 1
Standard Poll

Poll of 1000 people, 52% favor option A, 95% CI.

  • SE = sqrt(0.52×0.48/1000) = sqrt(0.0002496) = 0.01580
  • MOE = 1.96 × 0.01580 = 0.031 = 3.1%
  • CI: 52% ± 3.1% = (48.9%, 55.1%)
  • Includes 50%: result is within MOE of a tie
✓ MOE = ±3.1%
Example 2
Large Sample

Exit poll of 5000 voters, 99% CI, p=0.5.

  • SE = sqrt(0.25/5000) = 0.00707
  • MOE = 2.576 × 0.00707 = 0.0182 = 1.8%
  • 5× larger sample → MOE reduced by sqrt(5) ≈ 2.2x
✓ MOE = ±1.8% at 99% CI
Example 3
Small Survey

Internal company survey of 50 employees, 95%.

  • SE = sqrt(0.5×0.5/50) = 0.0707
  • MOE = 1.96 × 0.0707 = 13.9%
  • Very large MOE — small surveys are imprecise
  • Need n=400 for MOE ≤ ±5%
✓ MOE = ±13.9% (small sample is imprecise)

Real-World Applications

🗳️
Political Polling
Margin of error is always reported alongside poll results in reputable journalism.
💼
Market Research
Customer satisfaction surveys need MOE < 5% for actionable insights.
📊
Academic Research
Required for any sample-based study to quantify estimation precision.
📰
Journalism
Statistical literacy: readers can judge if a result is within the margin of error.

Common Mistakes to Avoid

⚠️
Confusing MOE with accuracy

MOE quantifies sampling error only. A biased sample (e.g., only online respondents) has systematic error the MOE doesn't capture.

⚠️
Not adjusting for non-binary responses

MOE for proportions uses p(1-p). For 5-point Likert scales, different formulas apply. Don't use proportion MOE for continuous scale data.

⚠️
Ignoring the confidence level

MOE without stating the confidence level is meaningless. Always pair them: ±3% at 95% CI. At 99% CI, the same poll has ±3.9% MOE.

Frequently Asked Questions

What is margin of error in polls? ▾
The ±X% reported alongside survey results. It means if you repeated the poll many times, 95% of polls would have results within X% of the true population value.
Why do most polls use ~1000 people? ▾
With n=1000 and p=0.5, 95% MOE = ±3.1%. Getting to ±2% requires n=2401 (4× more people for 2× more precision). The cost-benefit of going beyond ~1000 is usually poor.
What is sampling error vs non-sampling error? ▾
Sampling error: captured by MOE, comes from measuring only a sample. Non-sampling error: biased questions, non-response bias, measurement error — not captured by MOE and often larger in practice.
Does MOE apply to subgroups? ▾
If you poll 1000 total but analyze the 200 who identify as Group X, the MOE for Group X is much larger (based on n=200, not 1000). Subgroup analysis requires caution.
Why is MOE smallest at p=0.5? ▾
Wait — actually MOE is largest at p=0.5! p(1-p) is maximized at p=0.5. That's why we assume p=0.5 when unknown — to get the conservative (largest) MOE estimate.

Related Math Calculators

📊 Confidence Interval📊 Sample Size📊 P-Value📊 Standard Error📊 Z-score

Formula Explorer connections

Interpretation: This formula summarizes data, models uncertainty or supports inference about a population or random process. Assumption: The sampling design and distribution assumptions must match the data. Independence, sample size, outliers and measurement quality can materially affect interpretation.

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