Exact Binomial Test Calculator — Free & Precise

The Exact Binomial Test. Seven successes in twenty trials looks below a fifty percent benchmark. The exact binomial test asks how surprising that count is without a normal approximation. Record x successes and a fixed total n, then keep the benchmark p-zero separate from the observed proportion p-hat equals x over n. Under the null, the complete count X follows a binomial distribution. Every possible integer from zero through n receives an exact probability. For count k, multiply the combination count by p-zero to the k and one minus p-zero to the n minus k. A left-tailed test adds probabilities from zero through x. A right-tailed test adds probabilities from x through n, including the observation. For a two-sided test, Distri Scope adds every outcome no more probable than the observed count under the null, wherever it lies. Compare that tail-specific exact p-value with alpha. Reject when p is at most alpha; otherwise fail to reject without proving the benchmark true.

Exact arithmetic still needs independent binary trials, one stable success probability, a consistent success definition, and a sample representing the intended population. Fix n before observing outcomes. Stopping when the running result looks favorable changes the sampling process and invalidates the ordinary fixed-n test. Exact means no large-sample approximation. It does not mean assumption-free, and it cannot repair dependence, changing probabilities, or biased collection. Because X has only n plus one possible values, attainable p-values move in jumps. A tiny sample cannot provide smoothly adjustable evidence. Do not always double the smaller tail. That shortcut can disagree with probability-ranked two-sided testing when the null distribution is asymmetric. The reported interval is the equal-tailed Clopper Pearson interval. It is exact and conservative, but it is a different construction from the minimum-likelihood two-sided p-value. Cohen's h summarizes observed minus benchmark magnitude. Two observed groups need a two-sample method, while paired binary outcomes need a paired method.

Suppose a pilot fixes twenty independent trials, observes seven successes, and tests p-zero equals zero point five, two-sided, at alpha point zero five. The observed proportion is seven over twenty, or zero point three five: fifteen percentage points below the benchmark, before any inference. Under a fair benchmark, the probability of exactly seven successes is about zero point zero seven three nine. That single bar anchors the two-sided ranking. The inclusive left tail through seven totals about zero point one three one six. The opposite inclusive tail is much larger. Probability ranking selects counts zero through seven and thirteen through twenty. Their exact null masses sum to zero point two six three two. That p-value exceeds zero point zero five, so fail to reject. Seven successes are compatible with a fifty percent process, not proof of it. The ninety-five percent Clopper Pearson interval runs from about zero point one five four to zero point five nine two. Cohen's h is minus zero point three zero five.

Clopper Pearson coverage is at least the stated level, so the interval is often wider than an approximation. The width honestly reflects limited information. Edge counts remain valid. With zero successes the lower endpoint is zero; with all successes the upper endpoint is one. Validate whole counts with zero at most x at most n, require n from one through ten thousand, and keep p-zero strictly between zero and one. Choose the success label, benchmark, alternative, and alpha before inspection. Redefining success or switching the tail afterward changes the promised error rate. Clusters, repeated observations, changing probabilities, and sequential monitoring need design-aware methods. Exact tail summation alone does not protect against them. Report x, n, p-hat, p-zero, alternative, alpha, the two-sided convention, exact p-value, Clopper Pearson interval, effect size, and precise conclusion. Test one small-sample proportion, inspect every exact tail, and report the uncertainty free at Distri Scope dot com.