Category
Probability Distribution Calculators
A distribution is a model of how a quantity varies. Pick the one whose assumptions match your situation and everything else — probabilities, critical values, p-values — follows from it.
Choosing the right distribution
The choice is made by the shape of the question, not by the data.
Counting successes in a fixed number of independent tries with the same probability each time? That is the binomial distribution — ten coin flips, a hundred emails opened or not, a batch of parts passing inspection.
Counting events in an interval when they arrive independently at a steady average rate? That is the Poisson distribution — arrivals per hour, defects per metre, goals per match.
Measuring something that is the sum of many small independent influences? That is the normal distribution — heights, measurement errors, and, crucially, the average of almost any sample.
Working with a sample mean when the population standard deviation is unknown? The t distribution is the normal's heavier-tailed cousin, and the extra weight in its tails is exactly the price of having estimated σ from the data.
Working with sums of squared deviations — variances, contingency tables? That is the chi-square distribution.
The theorem that makes it all work
The central limit theorem says the mean of a sample is approximately normal no matter what shape the population has, provided the sample is not tiny and the population variance is finite. That is why the normal distribution appears in tests of data that are visibly not normal: the test is about the mean, and the mean behaves.
Probability before distributions
Underneath sit the counting rules: probability of single and combined events, permutations and combinations for how many arrangements exist, expected value for what a random quantity averages to, and Bayes' theorem for updating a belief when evidence arrives. Bayes in particular is worth the ten minutes: it is the reason a 99%-accurate test for a rare condition still produces mostly false positives.
The 9 calculators in this category
- Binomial Distribution CalculatorBinomial probability calculator: P(X = k), P(X ≤ k), P(X ≥ k) and the rest, with the mean, variance, standard deviation and the full distribution drawn as a bar chart.
- Probability CalculatorCompute the probability of a single event, of two events happening together or either happening, and of at least one occurrence across repeated independent trials.
- Permutations and Combinations CalculatorCompute permutations and combinations with or without repetition, exact for large n via log-gamma, with all four counting rules compared in one table.
- Poisson Distribution CalculatorPoisson probability calculator for events in a fixed interval: exact and cumulative probabilities, mean, variance and the distribution drawn as a bar chart.
- Normal Distribution CalculatorFind the probability below, above or between any values under a normal distribution, or invert it to find the value at a percentile.
- Expected Value CalculatorCompute the expected value of a discrete random variable from its outcomes and probabilities, with the variance, standard deviation and each outcome's contribution.
- T Distribution CalculatorStudent's t distribution calculator: tail probabilities from a t value, or critical values for any confidence level and degrees of freedom, with the curve drawn.
- Bayes' Theorem CalculatorCompute a posterior probability from a prior, a sensitivity and a false-positive rate, with a natural-frequency table that makes the base-rate effect obvious.
- Chi-Square Distribution CalculatorChi-square distribution calculator: upper-tail p-values from a χ² statistic, or critical values for any significance level and degrees of freedom, with the curve drawn.
Questions people ask
When can I use the normal approximation to the binomial?
The usual rule is np ≥ 10 and n(1 − p) ≥ 10 — both, not either. Below that the binomial is visibly skewed and the approximation misjudges the tails, which is exactly where p-values live. The binomial calculator computes the exact answer regardless, so there is rarely a reason to approximate.
What is the difference between the t and the normal distribution?
The t distribution has heavier tails, and how much heavier depends on the degrees of freedom. It is the correct distribution for a sample mean when σ is estimated from the data. At 30 degrees of freedom the difference is small; at 5 it is large; past about 100 it is negligible.
Is the Poisson distribution just the binomial with large n?
It is the limit of the binomial as n grows and p shrinks with np held constant at λ. That is why it fits rare events among many opportunities, and why its mean and variance are both λ.