Confidence Interval Calculator Calculators

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A confidence interval (CI) is a range of values calculated from sample data that, with a specified probability (confidence level), contains the true population parameter. A 95% CI means that if the same procedure were repeated many times, 95% of the calculated intervals would contain the true parameter value. Confidence intervals are more informative than p-values because they communicate both the direction and magnitude of an effect and the precision of estimation. They are reported alongside means in scientific publications, clinical trials, and survey results.

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Confidence Interval Formula

CI = x̄ ± z* × (s / √n) (for large n, using z-distribution)

CI = x̄ ± t* × (s / √n) (for small n, using t-distribution with df = n − 1)

x̄ = sample mean; s = sample standard deviation; n = sample size; z* = critical z-value (1.96 for 95%, 2.576 for 99%); t* = critical t-value from t-table at appropriate df.

Common Critical Values

  • 90% CI: z* = 1.645
  • 95% CI: z* = 1.960
  • 99% CI: z* = 2.576

Worked Example

n = 50 patients, x̄ = 142 mmHg, s = 18 mmHg. 95% CI: SEM = 18/√50 = 2.546; CI = 142 ± 1.96 × 2.546 = 142 ± 4.99 = [137.0, 147.0] mmHg. Interpretation: we are 95% confident the true population mean blood pressure is between 137 and 147 mmHg.

Interpreting Confidence Intervals

  • Narrow CI: precise estimate (large n or low variability)
  • Wide CI: imprecise estimate (small n or high variability)
  • If CI for a difference does not include 0: statistically significant at α = 1 − confidence level
  • CI does NOT mean 95% of data fall in the interval — it refers to the parameter estimate

Glossary

Confidence Interval (CI)
A range of values calculated from sample data that contains the true population parameter with a specified probability; 95% CI uses critical value z* = 1.96 (or t* for small n).
Margin of Error
The half-width of a confidence interval: z* × (s/√n); determines how far the CI extends above and below the sample estimate; decreases with larger n or lower variability.
Critical Value (z* or t*)
The multiplier in a CI formula corresponding to the desired confidence level; z* = 1.96 for 95% CI (large n); t* depends on degrees of freedom (n−1) for small n via the t-distribution.

Frequently Asked Questions

A confidence interval gives a range of plausible values for a population parameter (e.g., mean) based on sample data. A 95% CI means: if the study were repeated many times and a CI calculated each time, 95% of those intervals would contain the true parameter. It does NOT mean there is a 95% probability the true value is in this specific interval — either it is or it isn't. A 95% CI of [45, 55] for a mean says the estimate is 50 with a margin of error of ±5, and values between 45 and 55 are all plausible.

95% CI = x̄ ± 1.96 × (s/√n) for large samples (n ≥ 30). For small samples, replace 1.96 with the t-critical value at df = n − 1: 95% CI = x̄ ± t* × (s/√n). Example: x̄ = 25, s = 4, n = 16: SEM = 4/√16 = 1.0; t*(df = 15, 95%) = 2.131; CI = 25 ± 2.131 × 1.0 = [22.87, 27.13]. In Excel: CONFIDENCE.T(0.05, s, n) gives the margin of error for small samples.

For a difference between two means or a treatment effect: if the 95% CI includes zero, the difference is not statistically significant at α = 0.05 — zero is a plausible value for the true effect. If the CI excludes zero (entirely positive or entirely negative), the difference is significant at α = 0.05. This correspondence between CI and hypothesis test is exact: a 95% CI that excludes the null hypothesis value is equivalent to rejecting H₀ at α = 0.05 (two-tailed test).

A wider CI means less precision in estimating the true parameter — the data are consistent with a broad range of values. Width is determined by: sample size (larger n → narrower CI via √n denominator); variability (higher SD → wider CI); and confidence level (higher confidence % → wider CI). A very wide CI for a treatment effect (e.g., mean difference −10 to +30) means the data cannot rule out either no effect or a large positive or negative effect — the study was probably underpowered. CI width is directly linked to statistical power and sample size planning.