Confidence Interval Calculator

A confidence interval is a range of values that likely contains an unknown population parameter (like a mean or proportion). For example, a 95% confidence interval of [45, 55] for average height means we're 95% confident the true population average falls between 45 and 55. It consists of a point estimate (sample statistic) plus/minus a margin of error. The interval width depends on confidence level, sample size, and variability. Wider intervals = more confidence but less precision; narrower intervals = more precision but less confidence.

A 95% confidence interval means: if we repeated the study many times, about 95% of the calculated intervals would contain the true population parameter. It does NOT mean there's a 95% probability the true value is in this specific interval—the true value either is or isn't in the interval. Common interpretation: 'We are 95% confident that the true mean/proportion lies between [lower bound] and [upper bound].' The confidence level reflects our long-run accuracy, not the probability for any single interval.

Higher confidence levels produce wider intervals. At 90% confidence, z = 1.645; at 95%, z = 1.96; at 99%, z = 2.576. Example: For a sample mean of 50 with standard error 5: 90% CI = [41.8, 58.2] (width 16.4), 95% CI = [40.2, 59.8] (width 19.6), 99% CI = [37.1, 62.9] (width 25.8). There's a tradeoff: higher confidence = wider interval = less precision. Most research uses 95% as a balance between confidence and precision.

Use mean confidence intervals when your data is continuous/numerical (heights, weights, test scores, revenue). Formula: x̄ ± z(s/√n). Use proportion confidence intervals when counting successes/failures or yes/no outcomes (percentage who passed, approval ratings, defect rates). Formula: p̂ ± z√(p̂(1-p̂)/n). For proportions, ensure np̂ ≥ 10 and n(1-p̂) ≥ 10 for the normal approximation to be valid. For small samples or extreme proportions, use exact binomial methods.

Larger samples produce narrower, more precise intervals. The margin of error formula shows why: ME = z(s/√n). Doubling the sample size reduces the margin of error by √2 ≈ 1.41. General guidelines: For means, n ≥ 30 allows the Central Limit Theorem to apply. For proportions, ensure np̂ ≥ 10 and n(1-p̂) ≥ 10. To determine required n, rearrange: n = (z×s/ME)² for means, n = p̂(1-p̂)(z/ME)² for proportions. Surveys typically need 400+ respondents for ±5% margin of error at 95% confidence.