How do I calculate margin of error for a survey?

For proportions use MOE = z * sqrt(p*(1-p)/n). Use z = 1.96 for 95% confidence, p = 0.5 for maximum MOE, and n = your sample size. For example, n=1000 gives MOE = 1.96 * sqrt(0.25/1000) = 3.1%.

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

For a 3% margin of error at 95% confidence with p=0.5, you need approximately 1,068 respondents. Use the formula n = (z/MOE)^2 * p*(1-p) = (1.96/0.03)^2 * 0.25 = 1,068.

What does 95% confidence level mean?

A 95% confidence level means that if the survey were repeated 100 times with the same method, approximately 95 of the resulting confidence intervals would contain the true population value.

What z-score is used for 95% confidence?

The z-score for 95% confidence is 1.960. For 90% use 1.645, and for 99% use 2.576. These correspond to the area under the standard normal distribution curve.

Why is p=0.5 used as the default proportion?

Using p=0.5 maximizes p*(1-p), which maximizes the margin of error. This gives the most conservative (largest) sample size estimate, ensuring the actual MOE will be no larger than calculated regardless of the true proportion.

Does margin of error account for biased survey questions?

No. The margin of error formula only accounts for random sampling error. It does not capture bias from question wording, non-response, or sampling frame issues, which can be larger than the statistical margin of error.

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

The margin of error is the plus-or-minus value. The confidence interval is the full range: result minus MOE to result plus MOE. For example, if 60% approve with MOE of 3%, the confidence interval is 57% to 63%.

Margin of Error Calculator | Confidence Interval for Surveys

Target MOE	Required n
±1%	9,604
±2%	2,401
±3%	1,068
±5%	385

What Is Margin of Error?

The margin of error (MOE) quantifies the uncertainty in a survey or poll result. When a news report says a candidate has 52% support "plus or minus 3 points," that 3 points is the margin of error. It defines a range around the measured result within which the true population value is likely to fall, at a specified confidence level.

Margin of error is used in political polling, market research, product quality testing, medical studies, and any situation where a sample is used to draw conclusions about a larger population. Understanding it is essential for interpreting survey data correctly and for designing studies with sufficient precision.

Margin of Error Formula for Proportions

For survey proportions (yes/no, approve/disapprove), the formula is:

\[ \text{MOE} = z \times \sqrt{\frac{p(1-p)}{n}} \]

Where:

z = confidence level z-score (1.645 for 90%, 1.960 for 95%, 2.576 for 99%)
p = observed proportion (use 0.5 if unknown, gives maximum MOE)
n = sample size

Using p = 0.5 is standard practice when the true proportion is unknown, because it maximizes the margin of error and therefore gives the most conservative (safe) sample size estimate.

Margin of Error Formula for Means

When measuring a continuous variable (income, weight, temperature), the formula uses standard deviation:

\[ \text{MOE} = z \times \frac{\sigma}{\sqrt{n}} \]

This is equivalent to using the standard error of the mean multiplied by the z-score for the desired confidence level. The NIST Exploratory Data Analysis guidelines illustrate how descriptive statistics are applied across quality assurance, scientific research, and process monitoring in engineering settings.

How Sample Size Affects Margin of Error

The relationship between sample size and MOE is not linear. To cut MOE in half, sample size must be quadrupled. This is why polling organizations use samples of approximately 1,000 respondents: at n=1,000 with p=0.5 and 95% confidence, MOE is about 3.1%, which is generally accepted as sufficient for most national polls. Going to n=4,000 only reduces MOE to about 1.5%, at four times the cost.

Common Confidence Levels and Z-Scores

The confidence level determines the z-score used in the margin of error formula. Choosing the right level depends on the stakes: medical and safety research typically requires 99%, while political polling and market research use 95% as the standard.

Confidence Level	Z-Score	Interpretation	Common Use
80%	1.282	1 in 5 chance the true value lies outside the interval	Preliminary research, exploratory studies
90%	1.645	1 in 10 chance outside the interval	Business surveys, informal polls
95%	1.960	1 in 20 chance outside the interval	Standard for most research and polling
99%	2.576	1 in 100 chance outside the interval	Medical studies, safety-critical research
99.9%	3.291	1 in 1000 chance outside the interval	Quality control, high-stakes decisions

The 95% confidence level is the de facto standard in academic research and journalism because it balances precision with practicality. Increasing to 99% requires a substantially larger sample to achieve the same margin of error, roughly 1.73x more respondents for the same precision.

Margin of Error by Sample Size, Quick Reference (95% Confidence)

Per the NIST/SEMATECH e-Handbook, margin of error is a function of the critical value, the estimated proportion, and the sample size. The Qualtrics margin of error guide describes it as the single most misunderstood statistic in survey research, primarily because it only measures sampling error, not all sources of error in a survey.

The table below shows the margin of error for common sample sizes at 95% confidence, assuming p = 0.5 (maximum MOE, the most conservative estimate). Use this as a quick reference when designing surveys or interpreting published poll results.

Sample Size (n)	Margin of Error	Typical Use Case
50	±13.9%	Small pilot study
100	±9.8%	Classroom or small group survey
200	±6.9%	Local community survey
400	±4.9%	Regional poll
500	±4.4%	Standard academic survey
1,000	±3.1%	National poll (industry standard)
1,500	±2.5%	High-precision national poll
2,000	±2.2%	Large-scale market research
5,000	±1.4%	Government census sampling
10,000	±1.0%	Large epidemiological studies

These figures assume a simple random sample. Non-random sampling methods, convenience sampling, quota sampling, self-selection, can introduce systematic bias that no statistical formula corrects for, regardless of how large the sample is.

Worked Example: Calculating Margin of Error Step by Step

A political polling firm surveys 800 registered voters and finds that 47% intend to vote for Candidate A. What is the margin of error at 95% confidence?

Step 1 : Identify the values: n = 800, p = 0.47, z = 1.960 (95% confidence level)

Step 2 : Calculate the standard error: SE = √(p × (1 − p) / n) = √(0.47 × 0.53 / 800) = √(0.000311) = 0.01764

Step 3 : Multiply by the z-score: MOE = 1.960 × 0.01764 = 0.0346, or approximately ±3.5%

Interpretation: The true level of support in the voting population is likely between 43.5% and 50.5%. Because this range includes 50%, the race is statistically too close to call, the result is within the margin of error.

Unknown proportion: If the true proportion is unknown, use p = 0.5 for the most conservative MOE. With n = 800 and p = 0.5: MOE = 1.960 × √(0.25 / 800) = 1.960 × 0.01768 = ±3.46%, nearly identical because 0.47 is close to 0.5. The NIST/SEMATECH e-Handbook of Statistical Methods is the authoritative reference for precision limits and appropriate use cases of statistical estimators, and should be consulted for edge cases beyond this calculator's scope. When working with proportion-based surveys, compute the sample proportion first with our p-hat calculator before feeding it into the margin of error formula.

The Most Common Margin of Error Mistakes

As confirmed by the SurveyMonkey research methodology guide, the margin of error only measures sampling error, the natural imprecision from surveying a subset of a population. In practice, Reddit's r/statistics community flags two additional mistakes so often they have become standard teaching examples: treating a "statistical tie" (where the gap is smaller than the MOE) as meaningless, and confusing the 95% confidence level with a 95% probability that any single interval contains the true value.

Confusing MOE with bias. Margin of error only accounts for random sampling variation. It does not account for systematic bias. If a survey reaches only smartphone users, the MOE is still ±3%, but results may be skewed toward younger demographics. A small MOE does not mean a result is accurate, it only means the sampling was consistent.

Applying the overall MOE to subgroups. A national poll of 1,000 respondents has ±3.1% overall, but when broken down by age group (roughly n = 200 per group), the MOE per subgroup rises to ±6.9%. Cross-tabulation always increases uncertainty, and this is frequently overlooked in media reporting of demographic breakdowns.

Using the wrong confidence level. Many online calculators default to 95% without stating it clearly. If a sample size was designed at 90% confidence but results are reported as if at 95%, the actual margin of error is wider than reported. Always verify which confidence level was used before interpreting a result. In regression contexts, pair margin of error with our linear regression calculator to understand how prediction uncertainty scales with sample size.

Margin of Error Calculator