Margin of Error Calculator

ME = z* × SE SE = s / √n SE = √(p(1 − p) / n)

ME

The margin of error — half the width of the confidence interval, in the units of the estimate. It is a half-width rather than a width because the interval is written as estimate ± ME, so the same number is subtracted on one side and added on the other. Its units follow the estimate: percentage points for a proportion, and whatever the data was measured in for a mean

z*

The critical value, and in this mode it is always the t value taken at n − 1 degrees of freedom rather than the familiar 1.96. The difference is small next to the sample sizes surveys usually have, but it is not zero and it is largest exactly where the margin of error is largest: at n = 2 the t value is 12.7062, so a two-person sample whose standard deviation is 4 gets a margin of error of 35.94

s

The sample standard deviation, used only in the mean mode. On a survey this is rarely known in advance, which is why polls report proportions instead — for a yes/no question the variability follows from the percentage itself and nothing has to be estimated separately

p

The sample proportion, used only in the proportion mode and written as a fraction in the formula and as a percentage on the panel. Its contribution to the width is at its maximum at 50%: half the sample agreeing and half disagreeing is the most uncertain a binary question can be, which is why the margin of error quoted for a poll is usually the worst case at p = 0.5 rather than that poll's own figure

n

The sample size, appearing as √n. That square root is the reason doubling the sample does not halve the margin of error — it divides it by only 1.41, so halving the margin needs four times the sample. It is also the only factor a researcher controls, since p comes from the data and the confidence level is a statement about how careful the answer should be

Use it when you have already collected data and want to know how much of the reported figure is noise, or when you are reading a number someone else reported and want to see what it is made of. The two most useful things it shows are both on the panel rather than in the main result: the critical value, which reveals whether the t or the z was used and therefore whether the analyst accounted for estimating the spread, and the standard error, which makes the square-root law visible — quadruple the sample and this one halves. For planning a study rather than interpreting one, the sample size page is the better tool, because it turns the question around and asks how many responses are needed for a margin you have already decided is acceptable. Reading a survey result, remember that the margin covers only sampling error. It says nothing about who answered, whether the question was leading, or whether the sample was drawn in a way that let everyone in the population be chosen. Those problems do not shrink with n, and a large sample with a small margin of error is exactly how a biased poll comes to look authoritative.

1. The default: a mean estimated from 100 observations

   1. Standard error: s / √n = 12 / 10 = 1.2
   2. Critical value: the t value at 99 degrees of freedom, 1.9842 — not 1.9600
   3. Margin of error: 1.9842 × 1.2 = 2.3811
   4. The estimate itself is not an input, so the answer is a half-width rather than an interval

   The critical value is the row worth reading, because it is where a hand calculation usually goes wrong: 100 observations is a large sample, and reaching for 1.96 instead of 1.9842 changes the answer by 1.2%. That is small here and enormous at n = 2, where the same substitution would turn 35.9 points into 5.5 — a six-fold understatement of the uncertainty. The page prints the critical value precisely so that this step is checkable rather than hidden inside a multiplication.

2. A proportion from 1000 responses

   1. Proportion as a fraction: 50% = 0.5
   2. Standard error in percentage points: √(0.5 × 0.5 / 1000) × 100 = 1.5811
   3. Critical value: the normal value 1.9600, because the proportion mode uses z
   4. Margin of error: 1.96 × 1.5811 = 3.099 percentage points

   This is the shape of the figure newspapers quote, and three details are worth separating. The critical value is 1.9600 here while the mean example above used 1.9842, because a proportion's variability is fully determined by p and needs no degrees of freedom. The standard deviation field sits on the panel unused — in this mode the variability comes from the percentage, not from a separate estimate. And p = 50% is the worst case, so the number generalises: any question in that survey has a margin of error no larger than this one.

3. Nine observations put the margin of error right on the critical value

   1. Standard error: 3 / √9 = 1, exactly
   2. Critical value at 8 degrees of freedom: 2.306
   3. Margin of error: 2.306 × 1 = 2.306
   4. Two of the three rows print the same number, which is a coincidence of this choice of inputs rather than a rule

   The coincidence is worth seeing once so it is not mistaken for a relationship later: the standard error came out at exactly 1, which makes the multiplication a no-op and leaves the critical value visible as the answer. Nine observations and a standard deviation of three is a natural small-sample setup, and this is the same t value the confidence interval page produces at n = 9 — the two pages share the whole computation and differ only in whether a sample mean was supplied to shift it.

4. The same data read at 99% instead of 95%

   1. Standard error unchanged at 1.2 — nothing about the data moved
   2. Critical value at 99% with 99 degrees of freedom: 2.6264
   3. Margin of error: 2.6264 × 1.2 = 3.1517
   4. Compare with the 95% case: 2.3811, so the extra confidence costs 32% more width

   Only one of the three rows changes, and that is the point of printing all of them: the confidence level touches the critical value and nothing else, while the sample size touches the standard error and nothing else. The two knobs are independent, and confusing them is a common way to misread what was changed. The trade itself is worth naming — 99% confidence is not more accurate, it is more cautious. It buys a wider interval that covers the truth more often and pins the estimate down less tightly on any single occasion.

The margin of error covers one source of error only: the variation that comes from having drawn a sample rather than measured the whole population. It says nothing about bias, and the two are not interchangeable — a sample of a thousand volunteers from a website has a small margin of error and can be wrong by twenty points, because the people who chose to answer are not a random draw. It also assumes the observations are independent. Surveying several people in one household, or measuring the same subject twice, reduces the effective sample size far below the number of responses, and the formula has no way to know; the margin comes out too narrow. Both the mean and the proportion formulas are large-sample approximations: the proportion version in particular is unreliable when n·p or n·(1 − p) is small, which is what happens with rare categories — a 2% response rate in a sample of 100 should not be reported with a symmetric plus-or-minus at all. And a margin of error describes one estimate in isolation. It is the wrong tool for comparing two groups, where what matters is the margin of error of the difference, which is not the sum of the two individual margins unless the groups are independent.

What does a margin of error of 3% actually mean?

It means that if the same survey were repeated many times, about 95% of the resulting intervals — the reported figure plus or minus 3 points — would contain the true population value. It does not mean the true value is within 3 points with certainty, and it does not mean the survey is wrong by up to 3 points. Two consequences follow that are easy to miss. The true value is either inside the interval or outside it, so the 95% describes the procedure rather than this one result. And the margins compound when you compare groups: if two candidates sit 4 points apart in a poll with a 3-point margin each, the race is not "within the margin" in any simple sense — the difference has its own margin of error, which is larger than either one.

Why is the margin of error smaller when the sample is larger, but not proportionally?

Because the sample size enters the formula under a square root. Going from 100 to 400 responses halves the margin of error, and going from 400 to 1600 halves it again — each halving costs four times the sample. This is the single most consequential fact about survey design, and it explains why polls converge on a few hundred to a couple of thousand respondents: the first thousand buys most of the precision available, and the next thousand buys half as much. It is also why a margin of error cannot be improved by better analysis. The only lever is more data, and it operates on a square root.

Is a bigger margin of error worse?

It is more honest, not worse. A wider interval is more likely to contain the truth, so a study reporting ±5 points at 99% confidence is making a safer claim than one reporting ±3 at 90% — the two are not comparable until the confidence level is stated. What is genuinely worse is an unreported margin, or one quoted without its confidence level, because then the reader cannot tell how much weight the number carries. The other thing a wide margin signals is simply that the sample was small, which is information rather than a defect: it says the study cannot resolve fine differences, and a study that cannot resolve them should not be read as if it had.

Why does the page reject a proportion of 0% or 100%?

Because the formula collapses there. The standard error for a proportion is √(p(1 − p)/n), and at p = 0 or p = 1 the factor p(1 − p) is zero, so the margin of error comes out as exactly zero. Printing "60% ± 0 points" is a claim of certainty that no sample can support — it says the population value is 60%, when all the data shows is that no one in this sample fell outside the category. There are proper ways to handle a zero or full count, and they all involve adding something to the counts rather than trusting the empty cell; none of them is a plain margin of error, so the page declines rather than returning a number that would be read as certainty.

Does the margin of error account for a biased sample?

No, and this is the most important limit of the whole idea. The margin of error measures sampling variability — the part of the error that comes from looking at some of the population instead of all of it. Bias is a different kind of error, produced by the way the sample was selected or the questions were asked, and it does not shrink as the sample grows. A voluntary online poll of ten thousand people can have a margin of error of one percentage point and still be off by fifteen, because the people who answered are not a random draw from the population. Large samples make sampling error small; they make bias look more precise. Reading any reported margin, ask first how the sample was drawn.

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

The standard error is the raw spread of the estimate, and the margin of error is that spread multiplied by a critical value that depends on the confidence level. So the standard error is a property of the data alone, and the margin of error is a statement about how much confidence you want to place in it. For a 95% interval the multiplier is about two, which is why the two numbers are often within a factor of two of each other and why they get conflated. Both are printed on the panel for that reason: the standard error tells you how much information the sample carries, and the margin of error tells you what you have decided to claim with it.

• 1.3.5.6. Measures of Scale — e-Handbook of Statistical Methods (the standard deviation the standard error is built from, and the n − 1 denominator the degrees of freedom come from) — National Institute of Standards and Technology (NIST)
• 6.2 Using the Normal Distribution — Introductory Statistics 2e (the critical value that multiplies the standard error, and the tail areas a confidence level is split into) — OpenStax (Rice University)
• 1.3.6.1. What is a Probability Distribution — e-Handbook of Statistical Methods (the sampling distribution whose spread is the standard error, which is what the plus-or-minus sign is describing) — National Institute of Standards and Technology (NIST)