Confidence Interval Calculator - 無料オンライン電卓

Confidence intervals are the standard way statisticians express uncertainty in estimates. Every time you read a poll result ('Modi has 48% approval, margin of error ±3%') or a clinical study ('the drug reduces blood pressure by 12 mmHg, 95% CI [8, 16]'), you're seeing confidence intervals at work. They convert a point estimate (a single number) into a range of plausible values for the true population parameter. This calculator computes confidence intervals for the population mean given a sample mean, sample standard deviation, and sample size - the most common type of CI used in surveys, A/B tests, and scientific experiments.

For Indian Class 11-12 statistics and B.Tech/B.Com/B.Sc statistics courses, confidence intervals are part of the inferential statistics syllabus. JEE/NEET don't typically test CI directly, but CAT and other management entrance tests do include CI questions. In professional contexts, market researchers, A/B testers in product teams, manufacturing quality engineers, clinical researchers, and economists all use CIs daily. The 95% confidence level is the de facto standard - it balances precision and certainty for most applications.

Confidence interval formula: CI = x̄ ± z × (σ / √n), where x̄ is sample mean, σ is population SD (or sample SD for large samples), n is sample size, z is the critical value for the confidence level (1.96 for 95%, 2.58 for 99%, 1.645 for 90%). The (σ / √n) term is the standard error of the mean - it measures how much the sample mean varies across different samples drawn from the same population.

Interpretation (critical, often misunderstood): a 95% CI does NOT mean 'there's a 95% probability the true mean is in this interval.' It means 'if we repeated this sampling procedure many times, 95% of the resulting intervals would contain the true mean.' The true mean is a fixed (unknown) value; the interval varies with each sample. The 95% refers to the long-run frequency of intervals capturing the true value.

Sample size effect: margin of error shrinks with √n. To halve the margin, you need 4× the sample size. To quarter the margin, you need 16× the sample size. This square-root law is why polls of 1,000-2,000 voters give ±3% margins (sufficient for most political reporting), but polls of 10,000 give just ±1% - the marginal precision gain is rarely worth the 10× cost.