What is the margin of error?

The margin of error (MOE) quantifies the uncertainty in a survey estimate due to random sampling. If a poll reports 52% support with ±3% margin of error at 95% confidence, the true population value is estimated to fall between 49% and 55% in 95 out of 100 similarly conducted surveys. It does not account for non-sampling errors like bias, dishonest responses, or poor survey design.

How do you calculate the margin of error?

The margin of error formula is ME = z × √(p̂(1 − p̂) / n), where z is the critical value for your confidence level (1.645 for 90%, 1.96 for 95%, 2.576 for 99%), p̂ is the sample proportion (use 0.5 if unknown for maximum MOE), and n is the sample size. For example, with n = 1000 and p̂ = 0.5 at 95% confidence: ME = 1.96 × √(0.25/1000) = 1.96 × 0.01581 ≈ 0.031 or ±3.1%.

Why is 95% confidence so common?

The 95% confidence level became standard largely through the influence of statistician Ronald Fisher in the early 20th century, who proposed p < 0.05 as a conventional threshold for statistical significance. It represents a practical balance: it is strict enough to filter out many spurious results but not so strict that real effects become impossible to detect. Polls use 95% by convention so results are comparable across surveys.

How do you reduce the margin of error?

The most effective way is to increase the sample size — the MOE shrinks proportionally to 1/√n. Doubling the sample size reduces the MOE by about 29%. For example, going from n = 100 (MOE ≈ ±9.8%) to n = 400 (MOE ≈ ±4.9%) cuts it in half. Using a lower confidence level (e.g., 90% instead of 95%) also reduces MOE. Improving p̂ accuracy (if the true proportion is near 0 or 1, not 0.5) also slightly reduces it.

What sample size gives a ±3% margin of error?

At 95% confidence with p̂ = 0.5 (maximum MOE scenario): n = (1.96 / 0.03)² × 0.5 × 0.5 = 4268 × 0.25 ≈ 1,068 respondents. This is why many national polls target samples of 1,000–1,200 — they reliably achieve ±3% at 95% confidence. For a ±5% MOE: n = (1.96/0.05)² × 0.25 ≈ 385. For ±2%: n ≈ 2,401. The quadratic relationship means cutting MOE in half requires 4× the sample.

How does the margin of error relate to election polling?

In election polling, the margin of error is the key indicator of poll precision. A poll showing Candidate A at 51% vs Candidate B at 49% with a ±3% MOE means the race is effectively a statistical tie — both candidates' true support could overlap within the confidence interval. Polling averages (aggregators like FiveThirtyEight) effectively reduce MOE by combining many polls, similar to increasing sample size. Systematic bias (partisan response bias, likely voter models) can shift results by more than the stated MOE.

Why can a poll of 1,000 people represent an entire country?

Sample size determines the margin of error, not the fraction of the population sampled. Polling 1,000 people from a population of 300 million gives the same MOE as polling 1,000 from 10,000 (assuming random sampling). At n = 1,000 with p̂ = 0.5 and 95% confidence, MOE ≈ ±3.1% — regardless of population size. The catch is that the sample must be truly random and representative — which is the hard part of polling, not the math.

What is the difference between sampling error and margin of error?

Margin of error captures only sampling error — the variability that arises from measuring a random sample rather than the whole population. Non-sampling errors (question wording bias, refusal to respond, false answers, sampling frame errors) are not captured in the MOE and can be much larger. A poll can have a ±2% margin of error from sampling but still be systematically wrong by 5% due to non-response bias or poor sample selection.

Margin of Error Calculator — Survey Sample Size

Calculates the margin of error and confidence interval for survey results, or finds the required sample size for a desired margin of error.

How to Use the Margin of Error Calculator

This margin of error calculator has two modes: Calculate Margin of Error from a known sample size, or Find Required Sample Size for a target margin of error. Select your confidence level (90%, 95%, or 99%), enter the sample proportion p̂ (use 0.5 for the most conservative estimate), and enter either the sample size or desired MOE. The calculator returns the margin of error as both a decimal and percentage, plus the confidence interval bounds.

For formal hypothesis tests on survey proportions, combine this with our t-test calculator for significance testing.

Margin of Error Formula

The margin of error for a proportion uses the following formula:

ME = z × √(p̂(1 − p̂) / n)

Where:

z = critical value: 1.645 (90%), 1.96 (95%), 2.576 (99%)
p̂ = sample proportion (the observed percentage, expressed as a decimal)
n = sample size

The standard error is SE = √(p̂(1−p̂)/n), and the confidence interval is [p̂ − ME, p̂ + ME].

Solving for Required Sample Size

Rearranging the formula: n = (z / ME)² × p̂(1 − p̂). With p̂ = 0.5 (which maximizes n), this simplifies to n = (z / ME)² × 0.25. Always round up to the next integer.

AdvertisementResponsive Ad

Confidence Levels and Z-Values

The z-value comes from the standard normal distribution — it is the number of standard deviations that captures the desired percentage of the distribution:

90% confidence → z = 1.645 — you accept a 10% chance of missing the true value
95% confidence → z = 1.960 — the most common standard; 5% chance of error
99% confidence → z = 2.576 — very conservative; only 1% chance of error, but requires larger samples

Higher confidence requires a larger z, which increases the MOE for a given sample size — or requires a larger sample to maintain the same MOE.

Sample Size vs Margin of Error

The MOE shrinks as sample size grows, but with diminishing returns — the relationship is proportional to 1/√n:

n = 100 → MOE ≈ ±9.8% (at 95%, p̂ = 0.5)
n = 400 → MOE ≈ ±4.9%
n = 1,000 → MOE ≈ ±3.1%
n = 2,500 → MOE ≈ ±2.0%
n = 10,000 → MOE ≈ ±1.0%

Going from ±5% to ±2.5% requires 4× the sample size. This is why reducing MOE below ±2% is expensive — you need very large samples. National polls typically target 1,000–1,500 respondents for a ±3% MOE at 95% confidence. For related statistical calculations, see our standard deviation calculator.

AdvertisementResponsive Ad

Combinatorics and Sample Counting

When designing a survey, you may also need to count how many distinct samples of size n can be drawn from a population — this involves combinations, which use factorials (n!). Our factorial calculator computes n! for any non-negative integer, which is the building block for binomial coefficients and combination formulas used in sampling theory.

Limitations of the Margin of Error

The MOE only accounts for random sampling error. It does not capture:

Non-response bias — if certain groups are less likely to respond, results may not represent the population
Question wording effects — subtle phrasing differences can shift responses by several percentage points
Social desirability bias — respondents may answer in socially expected ways rather than honestly
Coverage bias — online polls exclude people without internet access; telephone polls exclude those without phones
Timing effects — opinions can shift rapidly; a poll conducted over two weeks may conflate different conditions

A poll with a small MOE but significant systematic bias is less accurate than a poll with a larger MOE but sound methodology.

Sources & References

Margin of error — Wikipedia
Confidence interval — Wikipedia
Survey methodology — Wikipedia

Margin of Error Calculator