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Confidence Interval Calculator

Calculate the confidence interval for a population mean using sample statistics. Determine the margin of error and upper/lower bounds at common confidence levels.

About this Calculator

Calculate the confidence interval for a population mean using sample statistics. Determine the margin of error and upper/lower bounds at common confidence levels.

Formula & Calculations

Formula

CI = x̄ ± z × (σ / √n); Margin of Error = z × (σ / √n)
Where:
  • x̄=Sample mean
  • σ=Sample standard deviation
  • n=Sample size
  • z=Z-score for the chosen confidence level (1.645/90%, 1.96/95%, 2.576/99%)
  • ME=Margin of error
  • CI=Confidence interval bounds (lower, upper)

Assumptions

  • The sample data is drawn from a normally distributed population.
  • For large sample sizes (n ≥ 30), the Central Limit Theorem ensures approximate normality.
  • Standard z-scores are used; for small samples, a t-distribution should be considered.
  • Confidence level represents the proportion of intervals that would contain the true population parameter.

Calculation Examples

Example 1

Inputs:x̄ = 100, σ = 15, n = 36, Confidence = 95%
Result:95% CI: [95.1, 104.9], Margin of Error = 4.9

z = 1.96. ME = 1.96 × (15 / √36) = 1.96 × 2.5 = 4.9. CI = 100 ± 4.9 = [95.1, 104.9].

Example 2

Inputs:x̄ = 65, σ = 8, n = 25, Confidence = 99%
Result:99% CI: [60.88, 69.12], Margin of Error = 4.12

z = 2.576. ME = 2.576 × (8 / √25) = 2.576 × 1.6 = 4.12.

Frequently Asked Questions

What does a 95% confidence interval actually mean?

A 95% confidence interval means that if you repeated your sampling process many times and calculated an interval each time, about 95% of those intervals would contain the true population parameter. It does not mean there is a 95% probability that the parameter lies within a specific interval.

When should I use a t-score instead of a z-score?

Use a t-score when the sample size is small (n < 30) and the population standard deviation is unknown. With large samples (n ≥ 30), the z-distribution and t-distribution produce nearly identical results.