acalculator

What is my margin of error?

Type the sample size, the confidence level and either the survey percentage or the standard deviation. The margin of error calculator gives the ± margin, the standard error and the critical value it used.

Your numbers

Margin of error for a
Margin of error (±)
3.09898

The margin of error is ±3.09898 percentage points at 95% confidence.

In words
±3.09898 percentage points
Standard error
1.58114
Critical value
1.95996
Interval low
46.901%
Interval high
53.099%

Margin of error (±): 3.09898. The margin of error is ±3.09898 percentage points at 95% confidence.

How to calculate

Computes the margin of error of a survey percentage, z × √(p(1 − p) ÷ n), or of a mean, z × σ ÷ √n or t × s ÷ √n, at any confidence level.

Example with the default inputs (Margin of error for a Percentage, Sample percentage (p̂) 50%, Sample size (n) 1,000, Confidence level 95%): The margin of error is ±3.09898 percentage points at 95% confidence.

Method: Percentage: z × √(p̂(1 − p̂) ÷ n); mean: z × σ ÷ √n, or t(n − 1) × s ÷ √n.

  • A simple random sample from a population much larger than the sample (no finite-population correction).
  • The percentage margin uses the normal approximation, which needs enough successes and failures (n p̂ and n(1 − p̂) of 10 or more is a common rule); it gives 0 at 0% and 100%.
  • The margin covers random sampling error only, not bias from who answered or how the question was asked.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. Margin of error for a Percentage, Sample percentage (p̂) 50%, Sample size (n) 1,000, Confidence level 95% gives Margin of error (±) 3.098975, Standard error 1.581139, Critical value 1.959964.Source: NIST/SEMATECH e-Handbook of Statistical Methods, §7.2.4.1 Confidence intervals (the normal approximation p̂ ± z₁₋α/₂ √(p̂(1 − p̂) ÷ n)), https://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm (retrieved 2026-10-02)
  2. Margin of error for a Percentage, Sample percentage (p̂) 84.2%, Sample size (n) 500, Confidence level 95% gives Margin of error (±) 3.197037.Source: OpenStax, Introductory Statistics 2e, §8.3 A Population Proportion, Example 8.10 (n = 500, p′ = 0.842, 95%: EBP = 0.032), https://openstax.org/books/introductory-statistics-2e/pages/8-3-a-population-proportion (retrieved 2026-10-02) (EBP rounded to 0.032, so tolerance 2e-2)
  3. Margin of error for a Mean, Standard deviation is Known σ (z), Standard deviation 3, Sample size (n) 36, Confidence level 90% gives Margin of error (±) 0.822427, Standard error 0.5.Source: OpenStax, Introductory Statistics 2e, §8.1 A Single Population Mean using the Normal Distribution, Example 8.2 (σ = 3, n = 36, 90%: EBM = 1.645 × 0.5 = 0.8225), https://openstax.org/books/introductory-statistics-2e/pages/8-1-a-single-population-mean-using-the-normal-distribution (retrieved 2026-10-02) (the book rounds z to 1.645; the exact z gives 0.822427)
  4. Margin of error for a Mean, Standard deviation is From the sample (t), Standard deviation 10, Sample size (n) 25, Confidence level 95% gives Margin of error (±) 4.127797, Critical value 2.063899.Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.2 Confidence Limits for the Mean (Ȳ ± t₁₋α/₂,N−1 s ÷ √N), https://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm (retrieved 2026-10-02)

How it works

With confidence level C, the tail area is α/2 = (1 − C) ÷ 2.

  • Percentage p̂ (as a share, 0 to 1) from a sample of n: standard error SE = √(p̂(1 − p̂) ÷ n); critical value z with an upper-tail area of α/2; margin = z × SE. The page shows SE and the margin in percentage points (× 100) and the interval p̂ ± margin in percent.
  • Mean, known σ: SE = σ ÷ √n; margin = z × SE.
  • Mean, sample s: SE = s ÷ √n; margin = t × SE, with t from Student’s t distribution with n − 1 degrees of freedom and an upper-tail area of α/2.

z and t are exact quantiles, not rounded table values, so 95% uses z = 1.959964, not 1.96. Above 1,000,000 degrees of freedom, t comes from the Cornish–Fisher series t = z + (z³ + z) ÷ (4ν) + (5z⁵ + 16z³ + 3z) ÷ (96ν²), which is exact to double precision there.

Rules

  • n is a whole number from 2 to 10¹²; the confidence level is from 50% to 99.9999%.
  • The percentage is from 0% to 100% (default 50%); at 0% or 100% the margin is 0, a known weakness of the normal approximation.
  • The standard deviation is from 0 to 10³⁰⁰.
  • The interval for a percentage is not cut at 0% or 100%.

Output format. The margin, standard error and critical value to 6 significant digits. “In words” reads “±3.09898 percentage points” for a percentage and “±0.822427” for a mean (6 significant digits).

Worked examples by hand

Poll of 1,000 at 50%, 95% confidence. SE = √(0.5 × 0.5 ÷ 1,000) = 0.0158114, or 1.58114 points. z = 1.959964. Margin = 1.959964 × 1.58114 = ±3.09898 percentage points.

OpenStax Example 8.10. p̂ = 0.842, n = 500, 95%: √(0.842 × 0.158 ÷ 500) = 0.0163117, × 1.959964 = 0.0319704, which the book rounds to 0.032.

OpenStax Example 8.2. σ = 3, n = 36, 90%: SE = 3 ÷ 6 = 0.5; z = 1.644854; margin = 0.822427 (the book’s 1.645 gives 0.8225).

Mean with a sample s. s = 10, n = 25, 95%: SE = 10 ÷ 5 = 2; t with 24 degrees of freedom = 2.063899; margin = 4.127797.

Other questions people ask

What is the margin of error?

Half the width of a confidence interval: the most the sample estimate is likely to differ from the true value, at the chosen confidence level, from random sampling alone. A poll result of 52% ± 3.1 points means the interval 48.9% to 55.1%.

How do I calculate the margin of error for a survey?

Multiply the critical z value by the standard error √(p̂(1 − p̂) ÷ n). For 1,000 people at 50% and 95% confidence: 1.96 × √(0.25 ÷ 1,000) = 0.031, so ±3.1 percentage points.

Why use 50% when I do not know the percentage?

p̂(1 − p̂) is largest at p̂ = 50%, so 50% gives the largest, most cautious margin. Survey planners use it before they have results; any other percentage gives a smaller margin for the same sample size.

How does sample size change the margin of error?

The margin shrinks with the square root of n. Four times as many people halve the margin: at 95% and 50%, 250 people give about ±6.2 points, 1,000 about ±3.1, and 4,000 about ±1.5.

What is the margin of error for a mean?

z × σ ÷ √n when the population standard deviation σ is known, or t × s ÷ √n with n − 1 degrees of freedom when you use the sample standard deviation s. OpenStax’s example: σ = 3, n = 36 and 90% give 1.645 × 0.5 = 0.8225.

Does the margin of error cover every kind of error?

No. It covers random sampling error only. Who was asked, who answered, and how the question was worded can bias a result in ways no margin of error shows.