Margin of Error Calculator

1. Pick proportion for a percentage result, like the share of voters who back a candidate. Pick mean for an average, like the mean wait time in minutes.
2. Enter the sample size n. For a proportion, enter the result p in percent. Use 50 if you are planning a survey and do not know it yet. For a mean, enter the sample standard deviation s.
3. Choose a confidence level. Most polls use 95%. If your sample is a big share of a small group, enter the population size N too.
4. Read the margin of error at the top. The cards show the confidence interval, the critical value, the standard error, and how many people you would need to cut the margin in half.

For example, a poll of 400 people with p = 50% at 95% confidence has a margin of error of plus or minus 4.90 points.

The margin of error is a critical value times a standard error. The critical value comes from the confidence level. The standard error measures how much a result bounces around from sample to sample.

For a Proportion

Here p is written as a decimal, so 50% is 0.5. For n = 1,000 at 95% confidence, the standard error is the square root of 0.25 / 1,000, which is 0.0158. Then 1.96 x 0.0158 = 0.0310, or 3.10 percentage points.

For a Mean

With n = 36 and s = 12, the standard error is 12 / 6 = 2. At 95% confidence, z gives 1.96 x 2 = 3.92. The t value for 35 degrees of freedom is 2.030, which gives 4.06. For samples under 30, the t answer is the more accurate one.

To turn a result into a z-score, try our z-score calculator.

The only part of the proportion formula that depends on the result is p(1 - p). This product peaks at 50%. As p moves toward 0% or 100%, it shrinks.

That is why pollsters report the margin at 50%. It is a safe upper limit for every question in the survey, even before the answers come in.

The sample size sits under a square root. So the margin of error falls with the square root of n, not with n itself. This is often called the square-root law.

Going from 400 to 1,600 people cuts the margin from 4.90 to 2.45 points. Each extra point of precision costs more people than the last. To plan a survey from a target margin, use our sample size calculator.

The basic formula assumes the population is huge compared with the sample. When you sample a large share of a small group, there is less left to be uncertain about. The finite population correction (FPC) shrinks the margin to reflect this.

Example: you survey 500 of a company's 2,000 staff. The plain margin at p = 50% and 95% is 4.38 points. The FPC is the square root of 1,500 / 1,999, which is 0.866. So the adjusted margin is 4.38 x 0.866 = 3.80 points.

A common rule is to apply the FPC when the sample is more than 5% of the population. For a national poll of 1,000 adults, the factor is about 0.99999, so it changes nothing.

The margin of error only measures random sampling error. It assumes a true random sample where everyone answers honestly. Real surveys face other problems that it does not include:

• Non-response bias. The people who answer may differ from those who do not.
• Coverage bias. Some groups may have no chance of being picked, such as people without internet access in an online poll.
• Question wording and honesty. How a question is asked, and whether people tell the truth, can shift results.
• Weighting. Adjusting a sample to match the population adds extra variation, called the design effect.

So treat the margin of error as the smallest likely error, not the full error. For the spread of your raw data itself, see our standard deviation calculator.

The margin of error is the plus or minus range around a survey result that comes from random sampling. If a poll finds 52% with a margin of error of 3 points, the true share in the whole population is likely between 49% and 55%.

It means that if the same poll were repeated many times with fresh random samples, about 95 out of 100 of the intervals built this way would contain the true value. It does not promise that this one poll is within 3 points.

The term p times (1 - p) is largest when p is 50%, so 50% gives the widest margin of error. Pollsters use it when they do not know the answer in advance, because the real margin can only be the same or smaller.

At 95% confidence and p = 50%, you need about 1,068 people from a large population. That comes from n = 1.96 squared times 0.25 divided by 0.03 squared, which is 1,067.1, rounded up.

Not directly. The margin of error for the gap between two candidates in the same poll is larger, up to about twice the margin for each candidate alone. A 2-point lead in a poll with a 3-point margin is not a clear lead.

Use t when you estimate a mean from a sample and do not know the population standard deviation, especially when n is under 30. The t value is larger than z for small samples, so the margin of error is wider. For large samples the two are almost the same.

Use it when your sample is a large share of the whole population, usually more than 5%. For example, surveying 500 of 2,000 employees cuts the margin of error by about 13%. For national polls the correction is so close to 1 that it can be ignored.

• Mercer, A. 5 Key Things to Know About the Margin of Error in Election Polls. Pew Research Center (sample size, subgroups, and the margin for a lead).
• American Association for Public Opinion Research (AAPOR). Margin of Sampling Error / Credibility Interval (what the margin of sampling error does and does not measure).
• NIST/SEMATECH e-Handbook of Statistical Methods, section 7.2.4.1: Confidence Intervals (normal approximation interval for a proportion).
• NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.5.2: Confidence Limits for the Mean (t-based interval for a mean).
• OpenStax. Introductory Statistics 2e, section 8.3: A Population Proportion (error bound for a proportion).
• OpenStax. Introductory Statistics 2e, section 8.2: A Single Population Mean Using the Student t Distribution (when to use t instead of z).