Distributions
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A probability distribution describes how likely each possible value is. Pick one below and drag its parameters to see how the shape changes. Scroll down for the history behind each distribution.
- Drag the purple handles under the curve to select a region and see its probability. The handles also respond to arrow keys. Hold Shift for bigger steps.
- Switch on "Show CDF" for the cumulative view: click anywhere on the curve to read off a quantile, and follow the dashed guides from the handles to the probability axis.
- The dashed Γ, B, ν, erf, γ, and Iₓ badges near the formula are definitions: click one to see what it means.
- Switch on "Show samples" to overlay a random sample, or "Compare", offered where a natural reference exists (t, Poisson, Binomial, Chi-squared, Cauchy), to overlay a related distribution.
- In the History section, click "Read a short history" on any related distribution to jump straight to it.
General form
With your values
The cumulative distribution function F(x) gives the probability that a random draw is ≤ x. It runs from 0 (certain not to happen) to 1 (certain to happen). Click the CDF curve to see the quantile and probability at any point. If part of a distribution's tail lies outside the plotted window (the heavy-tailed Cauchy is the extreme case), the plotted curve tops out just below 1.
History
Normal
First derived by Abraham de Moivre in 1733 as an approximation to the Binomial, then developed further by Laplace and later named for Carl Friedrich Gauss, who used it in 1809 to analyze errors in astronomical measurements. Today it models any quantity produced by many small, independent additive effects. The Central Limit Theorem explains why it appears across fields from test scores to manufacturing tolerances.
Read on Wikipediat
Student's t emerges when estimating a normal mean from a small sample. Gosset's solution for unknown variance.
Chi-squared
The sum of k independent squared normals follows a χ²(k) distribution: how squared normals connect normal theory to variance estimation.
F
An F-distribution is the ratio of two independent χ² variables, the test statistic of ANOVA.
Normal
First derived by Abraham de Moivre in 1733 as an approximation to the Binomial, then developed further by Laplace and later named for Carl Friedrich Gauss, who used it in 1809 to analyze errors in astronomical measurements. Today it models any quantity produced by many small, independent additive effects. The Central Limit Theorem explains why it appears across fields from test scores to manufacturing tolerances.
Read on Wikipediat
Student's t emerges when estimating a normal mean from a small sample. Gosset's solution for unknown variance.
Chi-squared
The sum of k independent squared normals follows a χ²(k) distribution: how squared normals connect normal theory to variance estimation.
F
An F-distribution is the ratio of two independent χ² variables, the test statistic of ANOVA.
Poisson
Introduced by Siméon Denis Poisson in 1837 to describe rare legal misjudgments in French courts, but became famous through Ladislaus Bortkiewicz's 1898 study of Prussian soldiers killed by horse kicks. Used today for rare, independent events over a fixed interval: server requests per second, radioactive decay counts, typos per page.
Read on WikipediaExponential
The waiting time between events in a Poisson process is exponentially distributed: the gap, not the count.
Gamma
The waiting time for k Poisson events follows a Gamma. The Exponential is the special case k=1.
Binomial
The Poisson approximates the Binomial for rare events (large n, small p, moderate np) with the same mean.
Binomial
Formalized by Jacob Bernoulli in the late 1600s, published posthumously in Ars Conjectandi (1713), to count successes across repeated independent trials. It models A/B test conversions, quality-control defect counts, and any yes/no outcome repeated a fixed number of times.
Read on WikipediaNormal
The Normal approximates the Binomial when n is large: de Moivre's original 1733 discovery that started modern probability.
Poisson
The Poisson approximates the Binomial for rare events: same expectation, simpler form on the right tail.
Beta
The Beta is conjugate to the Binomial: posterior Beta(α + x, β + n - x) makes Bayesian updating algebraic.
Exponential
Emerges as the waiting-time counterpart to the Poisson process (the gaps between Poisson-distributed events are exponentially distributed) and became central to queueing theory as Agner Krarup Erlang developed it in the early 1900s while modeling telephone exchange traffic. Used for reliability analysis (time-to-failure) and modeling wait times in queues.
Read on WikipediaGamma
The Gamma generalizes the Exponential from one event to k: the full waiting time, not just the next gap.
Poisson
The Exponential describes the gaps; the Poisson counts the events. The gaps and the counts are linked by the same Poisson process.
t
Derived by William Sealy Gosset in 1908 while working as a chemist at the Guinness brewery. He published under the pseudonym "Student" because Guinness barred employees from publishing research, fearing it would reveal trade secrets. It solved a real problem: how to run hypothesis tests on small samples when the true population variance is unknown. That problem is everywhere in science, which is why the t-test is one of the most widely used statistical procedures.
Read on WikipediaNormal
The t approaches the Normal as degrees of freedom increase. Beyond df≈30 they're nearly indistinguishable.
Chi-squared
A t variable is Z / √(χ²/df), a normal divided by its estimated standard deviation.
Gamma
A generalization of the Exponential (the waiting time for several such events in a row), built on Euler's 18th-century work on the Gamma function. Used to model waiting times for multiple events, rainfall totals, insurance claim sizes, and as a building block in Bayesian statistics.
Read on WikipediaExponential
The Exponential is a Gamma with shape=1. The Gamma is the general waiting-time distribution.
Chi-squared
The χ² is a Gamma with shape=df/2, scale=2. The Gamma family subsumes χ² as a special case.
Chi-squared
Introduced by Friedrich Robert Helmert in the 1870s and independently developed by Karl Pearson in 1900 for his goodness-of-fit test, one of the first formal statistical hypothesis tests. It tests whether observed categorical data matches an expected distribution, and builds confidence intervals for variance.
Read on WikipediaNormal
The sum of squared independent normals: the definition that connects normal observations to variance estimation.
Gamma
The χ² is a special case of the Gamma: same family, more flexible parametrization.
F
An F variable is the ratio of two independent χ²/df, the test statistic for comparing variances.
F
Named "F" by George Snedecor in the 1930s in honor of Ronald Fisher, who developed the underlying theory for comparing variances in the 1920s. Central to ANOVA (analysis of variance): used to test whether several group means differ, or whether one regression model fits significantly better than another.
Read on WikipediaChi-squared
An F variable is the ratio of two independent χ² variables, each divided by its degrees of freedom. Both of the F's parameters come from those two χ²s.
t
The square root of F(1, ν) is t(ν). Degrees of freedom (ν) count the independent data points available after estimating parameters. An F with 1 numerator df tests the same restriction as a t-test.
Uniform
The simplest possible distribution. Equal likelihood was the implicit assumption running through classical probability since Pascal and Fermat's 17th-century letters on gambling problems. It's the basis of pseudorandom number generation: every other distribution in this tool can be simulated by transforming draws from a Uniform.
Read on WikipediaBeta
Studied by Euler and Legendre through the Beta function in the 18th century, later adopted by Bayesian statisticians to represent uncertainty about an unknown probability. Used heavily in A/B testing to model uncertainty about a conversion rate, and anywhere the unknown quantity is itself a probability between 0 and 1.
Read on WikipediaUniform
Beta(1,1) = Uniform: the Beta generalizes the Uniform into a flexible family over [0, 1].
Binomial
The Beta is conjugate to the Binomial: the posterior is Beta(α + x, β + n - x), making sequential updating a counting exercise.
Cauchy
First analyzed by Siméon Denis Poisson in 1824; Cauchy's name only became attached during an 1853 academic dispute with Bienaymé — a classic case of Stigler's law of eponymy. It is the standard counterexample to the Law of Large Numbers, since its sample mean never converges no matter how much data you collect. Physicists call it the Lorentz or Breit–Wigner distribution, where it describes resonance peaks in spectroscopy and particle physics.
Read on WikipediaNormal
Both are symmetric and bell-shaped, but the Cauchy's tails are so heavy its mean doesn't exist: the counterexample.
t
The Cauchy is t with 1 degree of freedom: the t family spans from Cauchy (df=1) to Normal (df=∞).