What is margin of error?

Margin of error tells you how much survey results might differ from the true population value due to sampling. A ±3% margin means results could be up to 3 percentage points higher or lower than sample value. For 95% confidence: in 95% of repeated samples, result falls within this range.

Why use 50% proportion when unknown?

A 50% proportion produces the largest margin of error, giving you the most conservative (worst-case) estimate. Most poll reports use this assumption to give the largest credible margin. For proportions further from 50%, the actual margin is smaller.

How does sample size affect margin of error?

Margin of error decreases with √n. Doubling sample size reduces ME by about 30%. Quadrupling halves ME. From 100 → 400 → 1600 → 6400 → 25,600 → ME: 9.8 → 4.9 → 2.5 → 1.2 → 0.6%. Diminishing returns at large samples.

What's the difference between margin of error and total survey error?

Margin of error: random sampling error only (what calculator gives). Total error: includes non-response, coverage, measurement, and processing errors. Real surveys may have total error 2-3× the stated margin. Be cautious of pollsters reporting only margin without addressing other errors.

What confidence level should I use?

95% is the standard for most polls and research. 90% for less critical surveys. 99% for high-stakes decisions. Higher confidence = larger margin = need larger sample. 95% balances reasonable precision with practical sample sizes.

How do I reduce margin of error?

Increase sample size (most effective). Lower confidence level (95% vs 90%). Better sampling design (stratified vs simple random). Reduce non-response. Improve question design. The most cost-effective approach depends on context; sample size has consistent impact.

Is margin of error the same as confidence interval?

Closely related. Confidence interval = point estimate ± margin of error. ME is the "± part." So 52% ± 3% gives CI of [49%, 55%]. Margin of error and confidence interval are two views of the same uncertainty.

Margin of Error Calculator

Enter your sample size, population proportion (or use 50% for maximum margin), and confidence level to calculate the margin of error for a survey or poll.

The margin of error quantifies how much a survey or poll result might differ from the true population value. When you see "Candidate Smith leads by 5 points with a 3% margin of error at 95% confidence," it means the actual lead could be 2 to 8 points; in 95% of repeated surveys, the result would fall within ±3% of the reported lead.

This calculator computes the margin of error for a sample proportion, given the sample size, proportion, and confidence level. Use 50% proportion when unknown — this produces the largest (most conservative) margin of error, which is why poll reports typically quote it.

Margin of error depends on three factors: 1. **Sample size**: larger sample = smaller margin (inversely proportional to √n). 2. **Confidence level**: higher confidence = larger margin (95% needs more margin than 90%). 3. **Population proportion**: 50% gives largest margin; near 0 or 100% gives smaller.

Common practical values: 3% margin requires ~1100 respondents at 95% confidence. 5% margin needs ~385. Halving the margin of error quadruples required sample size. Most national polls aim for 2-3% margin, requiring 1000-2400 respondents.

Margin of error is the headline number on poll results, but full reporting should include sample size, confidence level, and whether the survey used random sampling. Non-random sampling can produce hidden biases far larger than the stated margin of error.

Inputs

Sample Size

Population Proportion (%)

Use 50% if unknown (worst-case scenario)

Confidence Level

Population Size (0 = infinite)

Leave as 0 for large populations

Results

Margin of Error

± 4.90%

Confidence Interval

45.10% to 54.90%

Standard Error

0.025000

Z-Value

1.960

Last updated: May 29, 2026

Formula

**Margin of error for proportion:** ME = z* × √(p(1-p)/n) Where: - **z***: critical z-value for confidence level - **p**: sample proportion (or population if known) - **n**: sample size **Critical z-values:** | Confidence | z* | |---|---| | 80% | 1.282 | | 90% | 1.645 | | 95% | 1.960 | | 99% | 2.576 | **Worked example: 1000 surveyed, 50% support, 95% confidence** ME = 1.96 × √(0.5 × 0.5 / 1000) = 1.96 × √0.00025 = 1.96 × 0.0158 = 0.031 = 3.1% So result is ±3.1% with 95% confidence. **Finite population correction:** For smaller populations: ME_corrected = ME × √((N - n) / (N - 1)) Where N = population size. For small samples from small populations, correction is meaningful. **Maximum margin (worst case):** When p = 0.5: ME = z* × 0.5/√n. This is the maximum possible margin for given n and confidence. **Sample size for desired margin:** n = (z*/ME)² × p(1-p) Maximum (p = 0.5): n = z*² / (4 × ME²) = (z*/(2×ME))² For 3% margin at 95% confidence: n = (1.96/0.06)² = 1067. **Margin of error decreases with sample size:** | n | ME (95% CI, p=0.5) | |---|---| | 100 | ±9.8% | | 250 | ±6.2% | | 400 | ±4.9% | | 600 | ±4.0% | | 1000 | ±3.1% | | 2000 | ±2.2% | | 5000 | ±1.4% | | 10000 | ±1.0% | | 50000 | ±0.4% | Note: each doubling of sample size cuts ME by √2 (about 30%). **Common polling examples:** - **Gallup**: typical sample 1000, ME ±3.1%. - **CBS News**: 1500-2000, ME ±2.4-2.8%. - **YouGov**: 1500-3000, ME ±2-2.8%. - **Reuters**: 1500-3000, ME ±2.2-2.6%. - **State polls**: often 600-1000, ME ±3-4%. - **Exit polls**: 5000-15000, ME ±1.5-2.5%. **Different from total error:** - **Margin of error**: random sampling error only. - **Total error**: includes non-response, coverage, social desirability, framing effects. Real polling error can be 5-10% even with 1000+ respondents. **Two-sided interpretation:** ME of 3% means: true value could be 3 percentage points above OR below sample value. Total range = sample ± ME. **Practical thresholds:** - **<1% ME**: very precise, large sample. - **1-3%**: typical major poll. - **3-5%**: moderate sample. - **5-10%**: small sample or low confidence. **For absolute numbers (not proportions):** ME = z* × σ/√n Where σ is population SD (or sample SD as estimate). This requires knowing distribution. **Different sample selection:** - **Simple random**: standard ME calculation. - **Stratified**: smaller ME possible if strata internally homogeneous. - **Cluster**: larger effective ME than simple calculation. - **Convenience**: ME calculation may be misleading.

How to use this calculator

Enter sample size.
Enter expected proportion (use 50% if unknown).
Select confidence level (95% standard).
Optionally enter population size for finite correction.
Calculator returns margin of error.
For survey planning: also use sample size calculator.

Worked examples

National political poll

**Scenario:** Poll 1000 voters. 52% support candidate. 95% confidence. **Calculation:** ME = 1.96 × √(0.52 × 0.48 / 1000) = 1.96 × √0.00025 = 1.96 × 0.0158 = 3.1%. **Result:** 52% ± 3.1%. True support between 48.9% and 55.1% with 95% confidence. Election outcome uncertain at this margin.

Customer satisfaction survey

**Scenario:** Survey 200 customers. 75% satisfied. 99% confidence. **Calculation:** ME = 2.576 × √(0.75 × 0.25 / 200) = 2.576 × √0.0009375 = 2.576 × 0.0306 = 7.9%. **Result:** 75% ± 7.9%. True satisfaction between 67% and 83% with 99% confidence. Need larger sample for tighter estimate or accept 95% confidence (smaller margin).

Small community survey

**Scenario:** Town of 5,000 residents. Survey 500. 60% support new park. 95% confidence. **Calculation:** ME without correction = 1.96 × √(0.6×0.4/500) = 1.96 × 0.0219 = 4.3%. Finite correction = √((5000-500)/(5000-1)) = √(0.9) = 0.949. Corrected ME = 4.3% × 0.949 = 4.1%. **Result:** 60% ± 4.1%. True support between 55.9% and 64.1%. Finite correction reduces margin slightly for small population.

When to use this calculator

**Use margin of error for:**

- **Poll reporting**: standard inclusion. - **Survey design**: planning required sample. - **Result interpretation**: assess precision. - **Comparison**: how confident in difference? - **Decision making**: when results are uncertain.

**Margin reduction strategies:**

| Strategy | Effect | Trade-off | |---|---|---| | Larger sample | Reduces ME by √2 per double | Higher cost | | Lower confidence | Smaller ME (90% vs 95%) | More uncertainty | | Stratified sampling | Smaller effective ME | More design | | Better question design | Reduces non-sampling error | Pre-testing time |

**Reporting standards:**

Industry standards (e.g., AAPOR): - Report sample size. - Report margin of error. - Report confidence level. - Specify whether random sampling used. - Describe response rate.

**Common errors:**

- Confusing margin of error with total error. - Computing ME for non-random samples. - Forgetting confidence level. - Comparing margins without same confidence. - Treating ME as "the only error."

**Total survey error:**

Margin of error is only one component: - **Sampling error**: random variation (this calculator). - **Coverage error**: missing some population. - **Non-response error**: people who don't answer. - **Measurement error**: question wording effects. - **Processing error**: data entry, coding.

Real-world total error often 5-10%+ even with stated 3% margin.

**Sample size for different needs:**

| ME goal | n at 95% (p=0.5) | |---|---| | 10% | 96 | | 5% | 385 | | 4% | 600 | | 3% | 1067 | | 2% | 2401 | | 1% | 9604 | | 0.5% | 38,416 |

Returns diminish: 4× more for halving ME.

**Confidence levels:**

| Confidence | Use case | |---|---| | 90% | Exploratory, small samples | | 95% | Standard research | | 99% | High-stakes decisions | | 99.9% | Very strict applications |

Higher confidence = larger margin, requires larger sample.

**Polling examples by source:**

- **National polls**: typically 1000-2000. - **Phone surveys**: 1000-1500. - **Online panels**: 1500-3000 (oversampled to combat self-selection). - **Exit polls**: 5000-20000. - **Tracking polls**: 500-1000 daily.

**Calculation tips:**

- Use p=0.5 for worst-case margin. - For known proportions, use that for tighter estimate. - Apply finite correction for small populations (n/N > 5%). - Always specify confidence level. - Include sample size in reporting.

**Software:**

- **Excel**: simple formula. - **R**: pollster package or manual. - **Python**: scipy.stats or statsmodels. - **SurveyMonkey**: built-in calculator.

**Beyond simple margin:**

- **Design effect**: cluster vs simple random. - **Effective sample size**: adjusted for design. - **Bayesian credible intervals**: alternative to frequentist margin. - **Bootstrapping**: empirical confidence intervals.

Common mistakes to avoid

Confusing margin of error with total survey error. ME is just sampling error.
Comparing polls with different confidence levels directly.
Treating non-random samples like random samples for ME.
Forgetting that 50% gives largest ME (worst case).
Reading ME as a fixed certainty instead of probability statement.
Computing without specifying confidence level.
Ignoring non-sampling errors that can dwarf ME.

Frequently Asked Questions

Sources & further reading

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