Confidence Interval Calculator

A confidence interval describes a procedure, not a single number. A 95 percent interval means that if sampling and calculation were repeated indefinitely, 95 percent of the intervals produced would contain the true parameter; it does not mean there is a 95 percent probability that this particular interval contains it. For a mean, the half-width is a critical value times the standard error s/sqrt(n), with z equal to 1.645 at 90 percent, 1.960 at 95 percent and 2.576 at 99 percent. When the population standard deviation is unknown and the sample small, Student's t distribution with n-1 degrees of freedom replaces z and widens the interval: 2.776 at 4 degrees of freedom and 2.262 at 9, converging toward 1.96 as n passes about 30. Because width falls with the square root of n, quadrupling the sample size only halves the interval. Proportions need more care. The textbook Wald interval, p-hat plus or minus z times sqrt(p-hat(1-p-hat)/n), has genuinely poor coverage: it collapses to zero width when no successes or no failures are observed, can extend below 0 or above 1, and oscillates below the nominal level even at moderate sample sizes. The Wilson score interval, obtained by inverting the score test rather than approximating it, stays inside the unit interval and holds coverage close to nominal for small n and extreme proportions, which is why it is the preferred default. Every interval here assumes genuine random sampling and, for means, approximate normality of the sampling distribution, which the central limit theorem usually supplies by n of 30 for mildly skewed data but not for heavy tails. All of them quantify sampling error only, and say nothing about measurement bias, selection bias or confounding.

The two halves of this calculation have separate origins. The small-sample correction came from William Sealy Gosset, a chemist working for the Guinness brewery in Dublin, who needed to draw conclusions from batches of only a handful of measurements. Publishing under the pseudonym Student in Biometrika in 1908, in a paper titled The Probable Error of a Mean, he derived the distribution of the sample mean scaled by the sample standard deviation. Ronald Fisher put the result on a rigorous footing in the early 1920s, introduced the modern t notation and the degrees-of-freedom framing, and circulated the tables that appear in Statistical Methods for Research Workers from 1925. The interval itself was formalised by Jerzy Neyman, who presented the idea to the Royal Statistical Society in 1934 and set it out fully in a 1937 paper in the Philosophical Transactions of the Royal Society, coining the terms confidence interval and confidence coefficient and deliberately framing them in long-run frequency terms to avoid the Bayesian priors he wished to escape. For proportions, Edwin Bidwell Wilson published his score interval in the Journal of the American Statistical Association in 1927, and Charles Clopper and Egon Pearson gave the exact binomial interval in 1934. Both were long neglected in favour of the simpler Wald formula until Alan Agresti and Brent Coull documented its coverage failures in The American Statistician in 1998, and Lawrence Brown, Tony Cai and Anirban DasGupta reinforced the case in Statistical Science in 2001.