Margin of Error Calculator
Calculate margin error instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
Formula
MOE = z * sqrt(p(1-p)/n) * FPC
Where MOE = Margin of Error, z = z-score for confidence level, p = sample proportion, n = sample size, and FPC = finite population correction factor sqrt((N-n)/(N-1)) applied when population N is known.
Worked Examples
Example 1: Political Poll with 1,000 Respondents
Problem: A national poll surveys 1,000 likely voters and finds 52% support a candidate. Calculate the margin of error at 95% confidence.
Solution: p = 0.52, n = 1,000, z = 1.960\nStandard Error = sqrt(0.52 x 0.48 / 1000) = sqrt(0.0002496) = 0.01580\nMargin of Error = 1.960 x 0.01580 = 0.03097 = 3.10%\nConfidence Interval = 52% +/- 3.10% = 48.90% to 55.10%
Result: Margin of Error: 3.10% | Confidence Interval: 48.90% to 55.10%
Example 2: Employee Satisfaction Survey (Finite Population)
Problem: A company with 500 employees surveys 200 of them. 70% report satisfaction. Find the margin of error at 95% confidence.
Solution: p = 0.70, n = 200, N = 500, z = 1.960\nStandard Error = sqrt(0.70 x 0.30 / 200) = 0.03240\nFPC = sqrt((500 - 200) / (500 - 1)) = sqrt(0.6012) = 0.7754\nAdjusted MOE = 1.960 x 0.03240 x 0.7754 = 4.92%\nInterval: 65.08% to 74.92%
Result: Margin of Error: 4.92% (FPC-adjusted) | Confidence Interval: 65.08% to 74.92%
Frequently Asked Questions
What is the margin of error and why does it matter?
The margin of error is a statistical measure that quantifies the amount of random sampling error in a survey or poll result. It tells you how much the results from your sample might differ from the true population value. For example, if a poll shows 55% support with a margin of error of plus or minus 3 percentage points, the true support in the population likely falls between 52% and 58%. This concept is fundamental to understanding the reliability of any survey-based research, political polling, or scientific study that uses sampling rather than measuring an entire population.
How does sample size affect the margin of error?
Sample size has an inverse square root relationship with the margin of error, meaning that to cut the margin of error in half, you need to quadruple your sample size. Going from 100 respondents to 400 respondents cuts the margin of error in half, while going from 400 to 1,600 cuts it in half again. This diminishing return is why most national polls use sample sizes of around 1,000 to 1,500 people, which provides a margin of error of roughly 2.5 to 3 percentage points. Beyond about 2,000 respondents, the cost of additional interviews typically outweighs the marginal improvement in precision.
Why does the proportion value affect the margin of error?
The proportion (or expected percentage) affects the margin of error because the variance of a binomial distribution is maximized when the proportion equals 50%. The formula includes the term p times (1 minus p), which reaches its maximum value of 0.25 when p equals 0.5. When the proportion moves toward 0% or 100%, the variance decreases and the margin of error shrinks. This is why researchers often use 50% as a conservative estimate when the true proportion is unknown, as it produces the largest (most conservative) margin of error. If you expect a proportion near 10% or 90%, you would need a smaller sample size than if you expected a 50/50 split.
How is margin of error different from standard error?
Standard error measures the standard deviation of the sampling distribution of a statistic, while the margin of error equals the standard error multiplied by the critical value (z-score) for your chosen confidence level. The standard error tells you the typical amount of variation you would expect in sample estimates if you repeatedly drew samples of the same size. The margin of error then scales this up based on how confident you want to be. At 95% confidence, the margin of error is about 1.96 times the standard error, while at 99% confidence it is about 2.576 times the standard error. Standard error decreases as sample size increases.
Can the margin of error be zero or negative?
The margin of error can never be negative because it is based on a square root, which always produces a non-negative value. However, it can theoretically approach zero in two scenarios: when you survey the entire population (a census), or when the sample size becomes extremely large. In practice, even with a census, measurement errors, non-response bias, and data processing errors mean that some uncertainty always exists. Additionally, the margin of error only accounts for random sampling error and does not capture systematic biases such as poorly worded questions, non-representative samples, or dishonest responses, which can introduce errors larger than the calculated margin.
What sample size do I need for a specific margin of error?
To determine the required sample size for a target margin of error, use the formula n equals z-squared times p times (1 minus p) divided by the margin of error squared. For a 95% confidence level with a 50% proportion, you need 385 respondents for a 5% margin, 1,068 for a 3% margin, 2,401 for a 2% margin, and 9,604 for a 1% margin. These numbers assume an infinite or very large population. If your population is small, you can apply the finite population correction to reduce the required sample size. Remember that these calculations assume simple random sampling; clustered or stratified designs may require adjustments.