Distribution calculator
This calculator works out cumulative probabilities (the CDF), upper-tail probabilities, the probability between two values, the probability outside two values, and the probability density (PDF) or probability mass (PMF) for nine distributions. It also runs backwards — give it a probability and it returns the value that cuts off that share of the distribution, which is what an inverse distribution calculator does.
The distributions it covers
Nine in all: the normal, binomial, Student's t, Poisson, chi-square, F, exponential, Weibull and uniform distributions. Pick one from the Distribution menu and the input boxes relabel themselves for its parameters.
Two of them have a dedicated calculator of their own, with more detail and worked examples:
Related tools:
What you can calculate
- P(X ≤ 𝑥) — the cumulative probability up to a value.
- P(X ≥ 𝑥) — the upper-tail probability.
- P(𝑥₁ ≤ X ≤ 𝑥₂) — the probability of landing between two values.
- P(X ≤ 𝑥₁) + P(X ≥ 𝑥₂) — the probability of landing outside two values.
- PDF or PMF — the density at a value, or for a discrete distribution the probability of exactly that value.
- Value from a probability — the inverse, in either tail, or both tails at once.
- Z score — how many standard deviations a value sits from the mean. Normal distribution only.
Less-than, or less-than-or-equal?
For a discrete distribution (binomial, Poisson) the Inequality switch chooses between “less than” and “less than or equal to”, and between “greater than” and “greater than or equal to”. It changes the answer, because a single whole number carries real probability. For a continuous distribution the switch is hidden, because P(X < 𝑥) and P(X ≤ 𝑥) are the same number.
What is a probability density function (PDF)?
The Probability Density Function (PDF), written f(x), applies to a continuous random variable. It describes the relative likelihood of a given value: you would expect to meet more values where the PDF is high than where it is low. To get the probability that the variable falls in a range, you find the area under the density curve over that range — the integral of the density. The probability of getting one exact value of a continuous random variable is zero.
What is a probability mass function (PMF)?
The Probability Mass Function (PMF) applies to discrete probability distributions. The PMF gives the probability of getting one specific value in the distribution. The PMF and the PDF both describe likelihood — one for a discrete distribution, the other for a continuous one.