Margin of error
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Margin of Error & Sample Size Calculator
Margin of error from z, p, and n, or the sample size that hits a target margin of error.
Page updated 2026-09-14.
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Solving margin of error from a sample size of 1,000
At a 95% confidence level (z = 1.96), an assumed proportion of 50%, and a sample size of 1,000, the margin of error works out to 3.10% -- meaning a poll result from this sample could reasonably be off by about plus or minus 3.1 percentage points from the true population value.
The formula is: MOE = z x sqrt(p(1-p) / n) = 1.96 x sqrt(0.5 x 0.5 / 1000) = 1.96 x sqrt(0.00025) = 1.96 x 0.01581 = 0.0310, or 3.10%. Using p = 0.5 (an assumed 50/50 split) produces the largest possible margin of error for a given sample size -- it's a conservative, 'worst case' assumption commonly used when the true proportion isn't known in advance.
This same formula can be solved in either direction -- given a sample size, find the margin of error (as done here), or given a target margin of error, solve backward for the required sample size, which is exactly the calculation pollsters and researchers run when planning how many respondents they need before starting a survey.
What this margin of error assumes about your survey
MOE = z x sqrt(p(1-p)/n). Simple random sample, not a clustered poll. This formula assumes a simple random sample -- every member of the population had an equal, independent chance of being selected. Real-world surveys using cluster sampling, stratified sampling, or any non-random selection method have a different, typically larger, true margin of error than this simple-random-sample formula would calculate.
The z-value (1.96 here) corresponds to a 95% confidence level, the most commonly used standard in polling and research -- a 90% or 99% confidence level would use a different z-value (1.645 or 2.576 respectively), producing a correspondingly different margin of error for the same sample size.
This margin of error applies specifically to the overall sample result -- if you're looking at a subgroup within that sample (like just one demographic segment), the effective sample size for that subgroup is smaller, which means the true margin of error for that specific subgroup result is larger than the topline figure calculated here.
Related statistical planning tools
For calculating standard deviation and variance from an actual raw dataset (a different but related statistical building block), the Standard Deviation & Variance Calculator is the relevant tool.
For understanding how unusual a specific value is relative to a known distribution, the Z-Score / Normal Distribution Calculator uses a related statistical concept.
Frequently Asked Questions (FAQ)
How is the 3.10% margin of error calculated?
MOE = z x sqrt(p(1-p) / n) = 1.96 x sqrt(0.5 x 0.5 / 1000) = 1.96 x sqrt(0.00025) = 1.96 x 0.01581 = 0.0310, or 3.10%.
Why use p = 0.5 if the actual proportion being measured is unknown?
Assuming p = 0.5 produces the maximum possible margin of error for a given sample size, since p(1-p) is maximized at p = 0.5. It's a conservative, worst-case assumption commonly used specifically because the true proportion isn't known in advance of a survey.
Does this formula apply to any type of survey methodology?
MOE = z x sqrt(p(1-p)/n). Simple random sample, not a clustered poll. No. It assumes a simple random sample, where every population member has an equal, independent chance of selection. Surveys using cluster or stratified sampling have a different, typically larger, true margin of error than this formula produces.
Why does the z-value change with confidence level?
Different confidence levels correspond to different points on the normal distribution -- 95% confidence uses z = 1.96, while 90% uses 1.645 and 99% uses 2.576. A higher confidence level requires a larger z-value, which increases the resulting margin of error for the same sample size.
Does this margin of error apply to a subgroup within my sample, or only the full sample?
Only the full sample as calculated. A specific subgroup within that sample has a smaller effective sample size, meaning its true margin of error is larger than the topline figure calculated for the full 1,000-person sample.
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