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Margin Of Error

Understand margin of error as confidence-interval half-width and how it changes with sample size, variability, and confidence level.

Referenceengineering & statisticsPublishedLast reviewed: 2026-05-16

Use margin of error when you need to state how close an estimate should be to the underlying mean or proportion, within a selected confidence method.

What This Means

Margin of error is the half-width of a two-sided interval. If a mean estimate is 12.4 +/- 0.3 mm, the margin of error is 0.3 mm.

For planning, a smaller margin of error requires more samples. Higher variability and higher confidence also require more samples when all else is unchanged.

Key Relationships

mean interval half-width = critical value * s / sqrt(n)
proportion planning half-width = z * sqrt(p (1 - p) / n)
  • n is sample size.
  • s or sigma represents continuous-response variability.
  • p is the planning proportion for binary outcomes.
  • The critical value is set by the confidence level and method.

Use This When

  • Planning how many continuous measurements are needed for a target mean estimate.
  • Planning how many binary inspections are needed for a target nonconformance-rate estimate.
  • Interpreting confidence interval output as estimate plus or minus a half-width.
  • Explaining the tradeoff between confidence, precision, and test effort.

Assumptions

  • The margin of error is absolute, not relative, unless the method explicitly says otherwise.
  • For means, margin of error uses the same units as the response and standard deviation.
  • For proportions, margin of error is an absolute proportion or percentage-point half-width.
  • The sample-size method matches the data type.

Limitations

  • Margin of error does not include systematic bias from measurement method, sampling plan, or process drift.
  • Narrow intervals from poor samples can still be misleading.
  • Very small samples may make normal-approximation planning weak.
  • Rare-event proportions may need exact/binomial, reliability, or acceptance-sampling methods.

Common Mistakes

  • Treating a four percentage point margin as four percent relative error.
  • Using a mean sample-size formula for pass/fail data.
  • Choosing a margin of error smaller than the measurement system can support.
  • Forgetting to round sample size up to the next whole observation.
  • Reducing sample size by lowering confidence without stating the decision risk.

Sources

This reference is based on the NIST/SEMATECH Engineering Statistics Handbook for confidence-interval and sample-size relationships, with Penn State statistics references used as corroboration for mean and proportion margin-of-error planning formulas.