Probability Distribution Calculator
Probabilities and quantiles for the normal, t, chi-square, F, exponential, binomial and Poisson distributions, accurate far into the tails.
Calculated in the browser from the incomplete gamma and beta functions, evaluated on whichever side of the distribution is small so far tails keep their precision.
How to use
- Pick a distribution and fill in its parameters — mean and standard deviation for the normal, degrees of freedom for t and chi-square, n and p for the binomial.
- To get a probability, choose the event (at most, less than, at least, more than, between, or exactly) and enter x, or a and b.
- To get a critical value, switch to x from probability, enter p, and choose the lower tail, the upper tail or a central interval.
- For counts, check whether you mean at least or more than: the tool treats the endpoint exactly as the wording says.
- Read the shaded chart to confirm you asked for the region you meant.
About this tool
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Choose a distribution, enter its parameters, and either give a value to get a probability — P(X ≤ x), P(X ≥ x), P(a ≤ X ≤ b) and the rest — or give a probability to get the value that cuts it off: a lower or upper tail, or a central interval. It covers the normal, Student's t, chi-square, F and exponential distributions and, for counts, the binomial and Poisson, and it replaces the printed tables at the back of a statistics book: the familiar 1.960 for 97.5% of the normal, 2.228 for t on 10 degrees of freedom, and 3.841 for a chi-square on 1 all come out as the tables print them. Two things make it more careful than a quick calculator. Every probability is computed from the incomplete gamma or beta function on the side of the distribution where it is small, never as one minus something close to one, so the chance that a standard normal exceeds 8.3 — 5.2 × 10⁻¹⁷ — keeps all its significant digits instead of rounding to zero; the same holds for a binomial count hundreds of standard deviations out. And for counts the endpoint is treated as it should be: for a binomial with n = 10 and p = 0.3, P(X ≥ 3) is 0.617 and P(X > 3) is 0.350, a difference that a formula written for continuous variables silently loses. Quantiles of a count follow the definition scipy and R use, the smallest value whose cumulative probability reaches the target, because there is usually no value that hits it exactly. Quantiles of continuous distributions are found to full precision even at probabilities like 10⁻¹²; checking them by feeding the answer back into scipy's own distribution functions showed them to be the more exact of the two where scipy's inverse drifts in the fifth digit. The chart shades the area being asked about. Mean and standard deviation are shown too, marked undefined where they do not exist, as for t with one degree of freedom.
Frequently asked questions
Why do P(X ≥ 3) and P(X > 3) differ?
Because a count can be exactly 3. For a binomial or Poisson variable, P(X ≥ 3) includes the probability of 3 and P(X > 3) does not; with n = 10 and p = 0.3 that single value carries 0.267 of the probability. For a continuous variable any single value has probability zero, so the two agree there.
How accurate are very small probabilities?
To about twelve significant digits, even in the far tails. Each tail is computed directly — the upper tail of the normal from the incomplete gamma function, the upper tail of a binomial from the incomplete beta function — rather than as one minus the lower tail, which would lose every digit once the answer drops below about 10⁻¹⁶. The values were compared with scipy across the range, down to probabilities around 10⁻²⁹³.
What is the quantile of a discrete distribution?
Usually no count has cumulative probability exactly p, so the convention is to return the smallest k whose cumulative probability is at least p (and, for the upper tail, the smallest k whose upper-tail probability is at most p). That is what scipy's ppf and isf and R's qbinom and qpois return, and the tool follows it.
Why is the mean undefined for some settings?
Because the integral that defines it does not converge. Student's t with one degree of freedom — the Cauchy distribution — has no mean, and its variance is infinite for two or fewer degrees of freedom; the F distribution needs more than two denominator degrees of freedom for a mean and more than four for a variance. The tool marks these rather than printing a meaningless number.
Which critical value do I need for a two-sided test?
For a two-sided test at 5%, use the central 95% interval, or equivalently the upper-tail value for p = 0.025 — which is where 1.960 and 2.228 come from. For a one-sided test at 5%, use the upper tail with p = 0.05, which gives 1.645 for the normal.
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