Poisson Distribution Calculator
Rare events at a steady rate — where variance equals the mean.
Work out Poisson Distribution. Rare events at a steady rate — where variance equals the mean. Correct where the obvious implementation is not.
Probability of exactly 2 events
22.4042%
mean 3 · sd 1.732
P(k) = λᵏe⁻ᵛ/k!, for events happening independently at a constant average rate. Calls to a helpdesk, arrivals at a queue, decays in a sample. The distinctive feature is that the variance equals the mean, so the standard deviation is √λ. Real count data that is far more spread out than that is over-dispersed, and Poisson is the wrong model for it — a common finding in practice and worth checking before trusting a fit. It is also the limit of the binomial as trials become many and the per-trial probability small, which is why it works for rare events in large populations.
How the Poisson Distribution Calculator works
Enter an average rate and a number of events for the exact probability and both tails. The variance of a Poisson distribution equals its mean, which is the quickest way to tell whether real count data actually fits it.
Also known as: poisson probability calculator · poisson formula calculator · probability of k events · arrival rate probability calculator
Frequently asked questions
What is the Poisson formula?
P(k) = λᵏe⁻ᵏ/k!, where λ is the average number of events per interval. It applies to events occurring independently at a constant average rate.
When should I use Poisson rather than binomial?
When you have a rate rather than a fixed number of trials — calls per hour, defects per metre, decays per second. Poisson is also the limit of the binomial when trials are many and per-trial probability is small.
How do I know if my data is Poisson?
Check whether the variance matches the mean. Real counts are often far more spread out than that — over-dispersed — in which case Poisson is the wrong model and a negative binomial usually fits better.
Can λ be a decimal?
Yes. It is an average rate, not a count, so 2.7 calls an hour is perfectly valid even though you can never observe 2.7 calls.
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