Sampling Error Calculator

Find the sampling error (margin of error) of a survey or poll from your sample proportion, sample size, and confidence level.

Quick Facts

Formula
MoE = z × √(p(1−p)/n)
z = 1.645 (90%), 1.96 (95%), 2.576 (99%). A finite population correction is applied when a population size is entered.

Your Results

Calculated
Margin of error
-
± percentage points
Confidence interval
-
For the proportion
Standard error
-
√(p(1−p)/n)
Critical z-value
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For chosen confidence

Ready

Enter your sample details and press Calculate.

What sampling error measures

Sampling error is the difference between a value estimated from a sample and the true value in the whole population. When you poll 1,000 people instead of asking everyone, the result you get is not exact — a different random 1,000 people would give a slightly different answer. The margin of error quantifies how far off your estimate is likely to be, purely because you sampled rather than measured everyone. It does not account for bad question wording, non-response, or a biased sample; it captures only the random variation inherent in sampling.

The formula

For an estimated proportion (a percentage such as "52% approve"), the margin of error is:

Margin of error = z × √(p(1 − p) / n)

where p is the sample proportion written as a decimal, n is the sample size, and z is the critical value from the standard normal distribution for your chosen confidence level. The quantity √(p(1 − p) / n) on its own is the standard error of the proportion; multiplying by z widens it into a confidence interval. When the sample is a large fraction of a finite population, multiply by the finite population correction factor √((N − n) / (N − 1)), where N is the population size, to shrink the error accordingly.

Critical z-values

  • 90% confidence: z = 1.645
  • 95% confidence: z = 1.96 (the standard for most published polls)
  • 99% confidence: z = 2.576

Why p = 50% is the "worst case"

The term p(1 − p) is largest when p = 0.5, giving 0.25. Because that maximizes the standard error, pollsters who don't yet know the result — or who want a single margin that covers every question on a survey — assume p = 50%. That is why headline margins of error are conservative: a proportion near 10% or 90% actually has a smaller sampling error than the reported figure.

Common reference points

At 95% confidence with p = 50%, a sample of n = 400 gives roughly ±4.9 points, n = 1,000 gives about ±3.1 points, and n = 2,401 gives about ±2.0 points. Because error shrinks with the square root of n, halving the margin requires roughly quadrupling the sample. This is why national polls rarely exceed a few thousand respondents — the accuracy gained per extra interview keeps falling.

Frequently Asked Questions

How is sampling error calculated?
For a proportion, sampling error (margin of error) = z × √(p(1 − p) / n). With p = 0.50, n = 1,000 and 95% confidence (z = 1.96), that is 1.96 × √(0.25 / 1000) ≈ 0.031, so ±3.1 percentage points. Enter a population size to apply the finite population correction for smaller populations.
What sample size do I need for a ±3% margin of error?
Solve n = z²·p(1 − p) / E². Using the worst case p = 0.5 at 95% confidence, n = 1.96²·0.25 / 0.03² ≈ 1,068. For ±5% you need about 385 and for ±1% about 9,604. Because error falls with √n, tightening the margin quickly gets expensive.
Does sampling error account for bias?
No. The margin of error only measures random variation from sampling. It says nothing about non-response bias, poorly worded questions, or an unrepresentative frame. A poll with a tiny margin of error can still be badly wrong if the sample itself is skewed.
When should I use the finite population correction?
Apply it when your sample is a meaningful fraction of the total population — a rule of thumb is when n exceeds about 5% of N. Surveying 300 of a 1,000-member club, for instance, gives a noticeably smaller error than the uncorrected formula suggests. For very large populations (a national electorate) the correction is negligible, so you can leave the population field blank.

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