📏 Margin of Error Calculator

Calculate the margin of error, confidence interval, and required sample size for different margin of error targets. Supports finite population correction.

Sample size (n)

Population size (optional, for finite correction)

Confidence level

Proportion estimate (p)

Use 0.5 for maximum (most conservative) MOE

What is Margin of Error?

Margin of error (MOE) is a statistical measure that quantifies the uncertainty in a survey or poll result due to random sampling. When a pollster surveys 1,000 people instead of an entire population of millions, the result is an estimate — not a precise measurement. The margin of error expresses how far the true population value is likely to be from the sample result, at a given confidence level. A poll showing "Candidate A at 52% ± 3%" means the true support is likely between 49% and 55%, with 95% confidence.

MOE is calculated as z × √(p(1−p)/n), where z is the critical value for the chosen confidence level (1.96 for 95%, 2.576 for 99%), p is the observed proportion, and n is the sample size. The most common convention in political polling is 95% confidence with p = 0.5 (which gives the largest possible MOE for any proportion — a conservative worst-case). This is why "±3 points" is standard for a poll of about 1,000 respondents: 1.96 × √(0.25/1000) ≈ 0.031.

Critically, margin of error only accounts for random sampling error — the chance variation that results from surveying a random subset of the population. It does not account for systematic errors such as non-response bias, question wording effects, undercoverage of certain demographic groups, or the Bradley effect in political polls. A technically correct MOE reported alongside a poorly designed survey can give a false sense of precision. Understanding MOE is essential for critically interpreting any survey-based claim in the media, business, or research.

Margin of Error Formulas

Margin of Error

MOE = z* × √(p(1−p)/n)

Finite Population Correction

MOE_fpc = MOE × √((N−n)/(N−1))

Required Sample Size

n = (z*/MOE)² × p(1−p)

How the Margin of Error Calculator Works

Formula, assumptions, and calculation steps for this statistics tool.

Methodology

Statistics calculators organize sample data, apply the selected descriptive or inferential formula, and report the statistic with interpretation.

Calculation Steps

Enter raw values or summary statistics.
Clean separators and count the sample size.
Apply the relevant statistic, probability, or confidence formula.
Display the result with context such as degrees of freedom, percentile, or strength.

Assumptions and Limits

Samples should be representative of the population being studied.
Normality or independence assumptions apply only where the selected method requires them.
Rounded results may differ slightly from spreadsheet software.

Frequently Asked Questions

What is margin of error?

The margin of error quantifies the maximum expected difference between the true population proportion and a sample estimate. A ±3% MOE with 95% confidence means: if you ran the survey 100 times, 95 of those would produce results within 3% of the true value.

Why use p = 0.5?

When the true proportion is unknown, p = 0.5 maximises p(1−p) = 0.25, giving the largest (most conservative) MOE. Any other value of p will produce a smaller MOE, so p = 0.5 guarantees your sample is large enough regardless of the outcome.

What is finite population correction?

The standard MOE formula assumes an infinite population. When sampling from a finite population of size N, the FPC factor √((N−n)/(N−1)) reduces the MOE, reflecting the fact that you are sampling a large fraction of the population.

What confidence level should I use?

The 95% confidence level is the most common in research and polling. A higher confidence level (99%) gives a wider interval (larger MOE) but more certainty. A lower level (90%) gives a narrower interval but less certainty.

How do I reduce the margin of error?

MOE decreases as sample size increases (roughly proportional to 1/√n). Increasing your sample from 400 to 1600 halves your MOE. Using a lower confidence level or a proportion estimate different from 0.5 also reduces MOE.

Real-World Applications

🗳️

Political Polling

A pre-election poll showing Candidate A at 51% ± 3% (95% CI) means the true support could be 48–54% — within the MOE, the race is statistically too close to call.

📊

Market Research

Survey 400 consumers about brand preference and calculate the MOE to determine whether a reported 58% preference is statistically distinguishable from 50/50 (no preference).

🔬

Clinical Trial Design

Calculate the required sample size to achieve a maximum MOE of ±2% at 95% confidence — determining how many patients must be enrolled to produce a statistically meaningful result.

📰

Media Literacy

Critically evaluate reported survey results — a poll claiming "63% support the policy" with n=200 has a MOE of ±7%, making the true range 56–70% — a much less precise claim.

🏫

Education Research

Measure the average test score improvement after an intervention in a sample of 100 students — the MOE quantifies how confident you can be that the observed improvement reflects a true population effect.

💼

Customer Satisfaction Surveys

An NPS survey of 500 customers returning a Net Promoter Score of 42 has a MOE that determines whether a change from 38 last quarter is statistically significant or within sampling noise.

Common Mistakes

Confusing MOE with confidence interval

The margin of error IS half the width of the confidence interval. If MOE = ±3%, the 95% CI is 6 percentage points wide. "MOE" and "CI half-width" are synonymous for a proportion — but CI can refer to either the full range or the half-width depending on context.

Assuming MOE accounts for all survey errors

MOE only accounts for random sampling error. Non-response bias (certain groups refusing to participate), question wording, and interviewer effects can introduce systematic errors far larger than the reported MOE.

Using 50% as p when the true proportion is known to differ

p = 0.5 gives the maximum (conservative) MOE. If prior research suggests p ≈ 0.2, using the actual p in the formula gives a smaller (more precise) MOE — and a smaller required sample size for the same precision.

Treating a result outside the MOE as "statistically significant"

MOE describes the uncertainty around a single estimate, not the significance of a difference between two estimates. Comparing two groups each with their own MOE requires a different calculation (comparing overlapping confidence intervals).

Not accounting for design effect in complex surveys

The standard MOE formula assumes simple random sampling. Stratified, clustered, or weighted survey designs have a design effect that typically inflates the effective MOE. Complex surveys require adjusted formulas.

Margin of Error by Sample Size (p=0.5, 95% CI)

Sample Size (n)	MOE (95%)	Typical Use
100	±9.8%	Small pilot study
400	±4.9%	Local survey
600	±4.0%	Regional poll
1,000	±3.1%	National poll (standard)
1,500	±2.5%	Large national survey
2,400	±2.0%	High-precision research

References

Cochran, W.G. Sampling Techniques. Wiley, 1977.
Lohr, Sharon. Sampling: Design and Analysis. CRC Press, 2021.
AAPOR. Standard Definitions: Final Dispositions of Case Codes and Outcome Rates. AAPOR, 2023.
Moore, D.S., McCabe, G.P., and Craig, B.A. Introduction to the Practice of Statistics. Freeman, 2017.
Pew Research Center. Explaining Our Polling Methodology. Pew Research, 2024.