Margin of Error Calculator

Find the margin of error for your survey or estimate the sample size you need.

📖 What is Margin of Error?

The margin of error (MOE) quantifies the uncertainty in a survey or statistical estimate. It tells you how much the sample result might differ from the true population value, at a specified confidence level. A survey result of "52% ± 3%" means the true proportion is estimated to lie between 49% and 55% with the stated confidence.

Margin of error is central to survey research, opinion polling, scientific experiments, and quality control. It depends on three factors: sample size (larger samples = smaller MOE), population variance (more heterogeneous populations = larger MOE), and confidence level (higher confidence = larger MOE because you need a wider interval to be more certain).

Understanding MOE is critical for interpreting surveys correctly. When two candidates are within each other's margin of error (e.g., Candidate A at 51% and Candidate B at 49% with MOE ±3%), the race is statistically too close to call from that poll alone - the difference is within the range of sampling variability.

📐 Formula

MOE = z* × √(p(1−p)/n)

z* = critical value for the confidence level (1.645 for 90%, 1.96 for 95%, 2.576 for 99%)

p = sample proportion (use 0.5 for maximum MOE when unknown)

n = sample size

For a mean (known σ): MOE = z* × σ/√n

Required sample size for target MOE: n = (z*/MOE)² × p(1−p)

Confidence interval: [p̂ − MOE, p̂ + MOE]

📖 How to Use This Calculator

Select Calculate MOE to find the margin of error for a given sample size, or Find Sample Size to determine how many respondents you need.

Enter the sample size (or target MOE), the expected proportion p (use 0.5 if you don't know - this gives the most conservative/largest MOE), and confidence level.

Click Calculate. The MOE, confidence interval around p, and the Z* critical value appear.

📝 Example Calculations

Example 1 - Political Poll

n = 1,000, p = 0.5, confidence = 95%. MOE = 1.96 × √(0.25/1000) = 1.96 × 0.01581 = ±3.10%

If the poll shows 54% support, the 95% CI is [50.9%, 57.1%].

MOE = ±3.10%

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Example 2 - Clinical Trial

n = 400, p = 0.72, confidence = 99%. MOE = 2.576 × √(0.72×0.28/400) = 2.576 × 0.02244 = ±5.78%

Observed response rate of 72% has 99% CI: [66.2%, 77.8%].

MOE = ±5.78%

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Example 3 - Required Sample Size

Target MOE = ±2%, p = 0.5, confidence = 95%. n = (1.96/0.02)² × 0.25 = 9604 × 0.25 = 2,401

You need at least 2,401 respondents for ±2% MOE at 95% confidence.

Required n = 2,401

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Example 4 - Known Proportion

Target MOE = ±3%, p = 0.15 (known from prior research), confidence = 95%. n = (1.96/0.03)² × 0.15×0.85 = 4268 × 0.1275 = 544

With known prior proportion, fewer respondents needed than with p = 0.5.

Required n = 544

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Example 5 - 90% Confidence Level

n = 600, p = 0.5, confidence = 90%. MOE = 1.645 × √(0.25/600) = 1.645 × 0.02041 = ±3.36%

Lower confidence = smaller MOE. Same sample gives ±3.10% at 95% and ±3.36% at 90%... wait, lower confidence should give smaller MOE. At 90%: ±3.36% vs 95%: 1.96×0.02041 = ±4.00%. Actually for n=600: 90%: ±3.36%, 95%: ±4.00%, confirming lower confidence = smaller MOE.

MOE = ±3.36% at 90% confidence

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❓ Frequently Asked Questions

What is the margin of error?+

The margin of error (MOE) is the maximum expected difference between the sample statistic and the true population parameter, at a given confidence level. For example, a poll showing 54% support with MOE ±3% means the true support is estimated to be between 51% and 57% with the specified confidence (e.g., 95%).

What does 95% confidence level mean?+

A 95% confidence level means that if you repeated the survey many times, about 95% of the resulting confidence intervals would contain the true population parameter. It does NOT mean there is a 95% probability that the true value falls in any one particular interval - the true value either is or isn't in the interval.

How does sample size affect margin of error?+

Margin of error is inversely proportional to √n. Doubling the sample size reduces MOE by a factor of √2 ≈ 1.41. To halve the MOE, you need 4× the sample size. This is why there are diminishing returns to increasing sample size - large samples are expensive but the improvement in precision gets smaller.

What sample size do I need for a ±3% margin of error?+

For a proportion near 0.5 at 95% confidence: n = (1.96/0.03)² × 0.25 ≈ 1068. For ±2%: n ≈ 2401. For ±1%: n ≈ 9604. This is why major national polls use samples of about 1,000 - they give ±3% MOE at 95% confidence.

What is the difference between margin of error and standard error?+

Standard error is the standard deviation of the sampling distribution: SE = √(p(1−p)/n). Margin of error is the critical value times the standard error: MOE = z* × SE. The critical value (z*) depends on the confidence level: 1.645 for 90%, 1.96 for 95%, 2.576 for 99%.

Does margin of error depend on population size?+

For populations much larger than the sample (usually true for polls of thousands), MOE barely depends on population size. For small populations, apply the finite population correction: MOE_corrected = MOE × √((N−n)/(N−1)) where N is population size. This calculator uses the standard formula without correction.

What is the margin of error in a survey?+

The margin of error (MOE) is the range within which the true population parameter likely falls, given your sample. A poll reporting 52% support with a 3% MOE means the true support is likely between 49% and 55%. MOE is calculated as: MOE = z * sqrt(p(1-p)/n), where z is the critical value (1.96 for 95% confidence), p is the estimated proportion, and n is sample size.

How do I reduce the margin of error in a survey?+

Margin of error = z x sqrt(p(1-p)/n). To halve the margin of error, quadruple the sample size. Other options: increase confidence level (reduces precision) or if you know p is far from 0.5, use that estimate - a proportion of 0.1 or 0.9 gives a smaller margin than assuming p = 0.5.

What is the difference between margin of error and confidence interval?+

The margin of error is half the width of a confidence interval. A 95% CI of (42%, 52%) has a margin of error of plus or minus 5% around the 47% point estimate. The confidence interval gives the full range; the margin of error gives the plus-minus range. Both convey the same information, just presented differently.

How can I reduce margin of error in my survey?+

The most direct way is to increase sample size - MOE is inversely proportional to the square root of n. To halve the MOE, you need 4x the sample size. You can also reduce it by lowering confidence level (e.g., from 95% to 90%), though this is rarely advisable. Stratified sampling and other probability sampling techniques can also improve precision without changing n.

Margin of Error Calculator

📖 What is Margin of Error?

📐 Formula

📖 How to Use This Calculator

📝 Example Calculations

❓ Frequently Asked Questions

🔗 Related Calculators

📌 Quick Tips