Margin of Error Calculator
Work out how far a survey result could be from the true value, given the sample size and confidence level.
Formula and sources updated · Data sources · 1View algorithm, constants and default inputs (JSON)
Margin of error (pp)
Dots show recalculated samples. Other inputs stay fixed.
About this calculator
The margin of error is the plus-or-minus figure you see next to poll results. It shrinks with the square root of the sample size, so quadrupling your sample only halves the error. The confidence level decides how often the stated range is expected to contain the true value. The reported proportion interval is clipped to 0–100%; the normal approximation does not include non-sampling error or survey-design effects.
Worked example
calculated with the default values below
Sample size 500 · Observed proportion 50 % · Confidence level 95% · Population size (0 if unknown) 0 Margin of error 4.38 pp · Lower bound 45.62 % · Upper bound 54.38 % · Standard error 2.236 pp · Critical value (z) 1.9600
Method and assumptions
ME = z × √(p(1 − p) ÷ n × (N − n) ÷ (N − 1)) × 100 Normal approximation for a simple random sample with finite population correction when N is known. The interval bounds are clipped to 0–100%; design and non-sampling errors are excluded. The margin of error is the plus-or-minus figure you see next to poll results. It shrinks with the square root of the sample size, so quadrupling your sample only halves the error. The confidence level decides how often the stated range is expected to contain the true value. The reported proportion interval is clipped to 0–100%; the normal approximation does not include non-sampling error or survey-design effects. The references provide context and limitations. They do not validate this page’s illustrative multipliers, lookup values or predictions.
Data sources
The references provide context and limitations. They do not validate this page’s illustrative multipliers, lookup values or predictions.
Disclaimer
These calculators give general mathematical results. They suit study, planning and estimation — verify anything that feeds a graded submission, a published figure or a financial commitment.
Implementation source
This record describes the implementation used by this page. It is not independent validation. External references, when available, support only the stated formula or definition; estimates and custom rules retain their own assumptions.
Formula and variables
ME = z × √(p(1 − p) ÷ n × (N − n) ÷ (N − 1)) × 100
- ME
- margin of error, percentage points
- z
- critical z-value
- p
- sample proportion, 0–1
- n
- sample size
- N
- population size
Normal approximation for a simple random sample with finite population correction when N is known. The interval bounds are clipped to 0–100%; design and non-sampling errors are excluded.
Frequently asked questions
What does this margin of error include?
It estimates sampling variation for a simple random sample using a normal approximation, with a finite-population correction when population size is entered. It does not cover biased sampling, nonresponse, question wording or complex survey design.
Which inputs and limits apply to Margin of Error Calculator?
Use the units shown beside each field. These are form limits, not recommended settings. Sample size: 2–10,000,000; Observed proportion: 0–100 %; Population size (0 if unknown): 0–2,000,000,000 ME = z × √(p(1 − p) ÷ n × (N − n) ÷ (N − 1)) × 100 Normal approximation for a simple random sample with finite population correction when N is known. The interval bounds are clipped to 0–100%; design and non-sampling errors are excluded. The margin of error is the plus-or-minus figure you see next to poll results. It shrinks with the square root of the sample size, so quadrupling your sample only halves the error. The confidence level decides how often the stated range is expected to contain the true value. The reported proportion interval is clipped to 0–100%; the normal approximation does not include non-sampling error or survey-design effects. The references provide context and limitations. They do not validate this page’s illustrative multipliers, lookup values or predictions.
What is a complete worked example for Sample size = 500; Observed proportion = 50 %?
Inputs: Sample size = 500; Observed proportion = 50 %; Confidence level = 95% %; Population size (0 if unknown) = 0. Calculation: ME = z × √(p(1 − p) ÷ n × (N − n) ÷ (N − 1)) × 100. Results: Margin of error = 4.38 pp; Lower bound = 45.62 %; Upper bound = 54.38 %; Standard error = 2.236 pp; Critical value (z) = 1.9600.