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Free Confidence Interval & Margin of Error Calculator

Estimate confidence ranges for sample proportions using standard confidence levels and sample sizes.

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Tool 01

Confidence Interval + Margin of Error Calculator

Estimate confidence ranges for survey and conversion proportions.

Confidence Interval + Margin of Error Calculator

Estimate confidence ranges for proportion-based samples like survey or conversion data.

Sample size

Successes

Confidence level

Result summary

Observed rate

45.00%

Margin of error

±2.81%

Lower bound

42.19%

Upper bound

47.81%

Required sample size for roughly ±5% precision at 95% confidence: 385

{
  "confidenceLevel": 95,
  "sampleSize": 1200,
  "successes": 540,
  "observedRate": 0.45,
  "marginOfError": 0.02814835696803634,
  "interval": [
    0.42185164303196365,
    0.47814835696803637
  ],
  "standardError": 0.014361406616345072
}

How it works

Confidence Interval + Margin of Error Calculator: methodology and worked example

How this tool computes its result

Computes a Wald confidence interval for a single proportion: p̂ = successes / sample size, standard error = √(p̂(1−p̂)/n), margin = z × standard error, with the interval clamped to [0,1]. The z-score comes from a fixed lookup table for four confidence levels (80/90/95/99%). It also reports the classic minimum sample size needed for a ±5% margin at the selected confidence level, computed as ceil(z² × 0.25 / 0.05²), where 0.25 is the worst-case p(1−p) value at p = 0.5.

Worked example

With the tool's defaults — sample size 1,200, successes 540, confidence 95% — p̂ = 540/1200 = 0.45, z = 1.96, standard error = √(0.45 × 0.55 / 1200) ≈ 0.01436, and margin = 1.96 × 0.01436 ≈ 2.82%. The reported interval is roughly 42.2% to 47.8%. The "required sample for ±5%" figure computes ceil(1.96² × 0.25 / 0.05²) = ceil(384.16) = 385 — the well-known survey-sizing benchmark for ±5% margin at 95% confidence.

When not to use this tool

This only implements the Wald (normal-approximation) interval for a single binomial proportion at four preset confidence levels — it has no interval for means/continuous data (which would need a t-distribution), no two-sample comparison, and the Wald approximation itself is known to perform poorly for very small samples or proportions near 0 or 1.

Common mistakes

- Entering a success count greater than the sample size (e.g. 150 successes out of 100) — this is explicitly validated and blocked with "Success count must be between 0 and sample size." rather than silently computing a rate above 100%.
- Trusting the interval at small sample sizes or extreme proportions — the tool always uses the normal (Wald) approximation, never switching to an exact method like Wilson or Clopper-Pearson, so intervals can be misleadingly narrow when the sample is small or p̂ is near 0 or 1.
- Assuming the "required sample for ±5%" number reflects your actual observed rate — it always uses the conservative worst-case p = 0.5, so if your true proportion is far from 50%, the real number of respondents needed for ±5% may be smaller than the 385 figure shown.

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