Margin of Error Calculator

What the margin of error tells you

The margin of error tells you how far your sample result is likely to be from the true value for the whole population. When a poll reports that 60% of people support a policy “with a margin of error of plus or minus 3 points,” it means the real figure for the entire population is very likely between 57% and 63%. The margin of error is the bridge between the sample you actually measured and the population you want to describe.

This margin of error calculator works for two common situations: percentages or proportions (the share of people who chose an option) and means or averages (such as average age, income, or test score). Choose the matching tab, enter your numbers, and the result updates as you type.

How to use this calculator

Pick your confidence level. This is how sure you want to be that the true value falls inside your range. Most survey research uses 95%, which is the default. A higher confidence level produces a wider margin, because demanding more certainty means accepting a larger range.

Enter your sample size. This is the number of people or items you actually measured, not the size of the whole population. Larger samples produce smaller margins of error, but the benefit slows down as the sample grows, which the line chart in the tool makes clear.

Enter your sample proportion or standard deviation. For a percentage, set the slider to the share who gave the response you care about. For a mean, enter the standard deviation, which measures how spread out your data is. If you do not yet know your proportion, leave it at 50%, the value that produces the largest, most conservative margin of error.

Apply the finite population correction if needed. Tick this option only when you have surveyed a large share of a small, known population, for example 300 employees out of a company of 500. For large populations the correction makes almost no difference and can be left off.

The formulas behind the tool

For a proportion, the margin of error is the critical z-value multiplied by the standard error of the proportion: MoE = z times the square root of p times (1 minus p) divided by n. Here p is the sample proportion, n is the sample size, and z is the critical value for your confidence level.

For a mean, the margin of error is the critical z-value multiplied by the standard error of the mean: MoE = z times s divided by the square root of n, where s is the standard deviation.

When the finite population correction applies, both results are multiplied by the square root of (N minus n) divided by (N minus 1), where N is the population size. The critical z-values are 1.645 for 90% confidence, 1.96 for 95%, and 2.576 for 99%.

A worked example

Suppose you surveyed 1,000 people and 60% chose a particular option, and you want 95% confidence. The standard error is the square root of 0.6 times 0.4 divided by 1,000, which is about 0.0155. Multiplying by 1.96 gives a margin of error of about 3.0 percentage points. So you can be 95% confident that the true figure for the whole population is between roughly 57% and 63%.

Things to keep in mind

The margin of error only captures sampling error, the random variation that comes from measuring a sample instead of everyone. It does not account for biased questions, poor sampling methods, or people who decline to answer. A small margin of error on a badly designed survey can still be misleading.

The margin of error is largest when a proportion is near 50% and shrinks as it approaches 0% or 100%. For very small samples where the population standard deviation is unknown, a t-distribution gives a slightly wider and more accurate interval than the z-value used here.

Frequently asked questions

What is a good margin of error? Many surveys aim for plus or minus 5% or lower. Plus or minus 3% is common for national polls and typically requires a sample of around 1,000 respondents.

Does a bigger population need a bigger sample? Surprisingly, no. Beyond a few thousand, population size barely affects the required sample. A sample of about 1,000 gives a margin of error near 3% whether the population is 100,000 or 100 million.

How do I reduce my margin of error? Increase your sample size or accept a lower confidence level. Because the sample size sits under a square root, cutting the margin of error in half requires roughly four times as many responses.