Confidence Interval Calculator

A confidence interval (CI) is a range of values constructed from sample data that is likely to contain the true population parameter. A 95% CI means that if the study were repeated 100 times with different samples, approximately 95 of the 100 resulting intervals would contain the true population mean or proportion. It is NOT the probability that the true parameter lies in any single interval - once computed, the interval either contains the parameter or it does not.

CI = x̄ ± z* × (σ / √n), where x̄ is the sample mean, z* is the critical value for your confidence level (1.96 for 95%), σ is the population standard deviation, and n is the sample size. The term z* × (σ / √n) is the margin of error. When σ is unknown (the usual case), replace z* with the t* critical value from the t-distribution with n−1 degrees of freedom.

The critical value for a 95% confidence interval is z* = 1.96 (from the standard normal distribution). This means 95% of the area under the normal curve falls within ±1.96 standard deviations of the mean. For 90% CI, z* = 1.645; for 99% CI, z* = 2.576. These values come from the inverse of the normal CDF at (1 + confidence level) / 2.

CI = p̂ ± z* × √(p̂(1−p̂)/n), where p̂ is the sample proportion (successes/n), z* is the critical value, and n is the sample size. For example, if 240 of 400 people (p̂ = 0.60) prefer a product at 95% CI: margin = 1.96 × √(0.60 × 0.40 / 400) = 1.96 × 0.0245 = 0.048. The CI is [0.552, 0.648], or roughly 55.2% to 64.8%.

Being 95% confident means the procedure used to construct the interval captures the true parameter 95% of the time across repeated samples. It describes the reliability of the method, not the probability of any single interval. Once you calculate a specific interval like [4.2, 5.8], that interval either contains the true mean or it doesn't - there is no probability attached to a fixed, computed interval.

The margin of error is proportional to 1/√n. To halve the margin of error, you need 4 times the sample size. To reduce it by one-third, you need 9 times as many observations. This is why large-scale surveys (n = 1,000+) can achieve margins of error below ±3%, while small studies (n = 25) may have margins of ±20% or more at the same confidence level.

Use the z-distribution (standard normal) when: (1) the population standard deviation σ is known, or (2) the sample size n ≥ 30 (by the Central Limit Theorem, the sampling distribution is approximately normal). Use the t-distribution when σ is unknown and n < 30. The t-distribution has heavier tails, producing wider intervals that account for the additional uncertainty from estimating σ using the sample standard deviation s.

The margin of error (MoE) is the half-width of the confidence interval: MoE = z* × SE, where SE is the standard error (σ/√n for means, √(p̂(1−p̂)/n) for proportions). If a poll reports a candidate at 48% ± 3%, the 3% is the margin of error at whatever confidence level the poll used (typically 95%). The full CI would then be [45%, 51%].

Higher confidence levels produce wider intervals. At 90% CI, z* = 1.645; at 95% CI, z* = 1.96; at 99% CI, z* = 2.576. A 99% CI is about 31% wider than a 95% CI (ratio = 2.576/1.96 ≈ 1.31). The tradeoff is precision vs. certainty: a 99% CI is more likely to contain the true parameter but provides a less precise estimate of its location.

A confidence interval estimates where the true population mean lies. A prediction interval estimates where a single future observation will fall - always wider than the CI. For example, a CI might say the average height of men is [175.2 cm, 176.8 cm], while a prediction interval for one man's height might be [158 cm, 194 cm]. Use CI to estimate parameters; use prediction intervals to forecast individual observations.

Rearranging the margin of error formula: n = (z* × σ / MoE)². For proportions: n = z*² × p̂(1−p̂) / MoE². To achieve a 3% margin of error at 95% confidence for a proportion, using p̂ = 0.5 (worst case): n = 1.96² × 0.25 / 0.03² = 3.8416 × 0.25 / 0.0009 ≈ 1,068 observations.