Poisson Distribution Calculator
Poisson probabilities: exactly, at most or at least k events.
About this tool
Enter the average number of events per interval (λ) and the number of events k. The tool gives P(X = k) and the cumulative probabilities, the mean, variance and standard deviation, and a bar chart of the whole distribution.
When to use the Poisson distribution
The Poisson distribution describes counts of independent events in a fixed interval of time or space when they happen at a steady average rate λ: emails per hour, calls to a support line, defects per metre of cable, goals per match. The probability of exactly k events is P(X = k) = e^(−λ) λ^k ÷ k!, and cumulative probabilities add these up. With λ = 3 (say, three customers every ten minutes), the chance of exactly two in the next ten minutes is 22.4%, two or fewer 42.3%, and at least one 95.0%. A Poisson distribution's mean and variance are both λ, so its standard deviation is √λ. It also approximates the binomial distribution when there are many trials, each with a small chance of success. Honest limits: the model assumes events are independent and the rate is constant; real data often has more variance than λ (overdispersion) when events cluster or rates change through the day, and then a negative binomial model fits better. The probability calculator and the z-score calculator are related.
Frequently asked questions
What is the Poisson formula?
P(X = k) = e^(−λ) × λ^k ÷ k!, where λ is the average number of events per interval.
What is λ in a Poisson distribution?
The average number of events in the interval. It is both the mean and the variance.
When should I use Poisson instead of binomial?
For counts of events in time or space with no fixed number of trials, or as an approximation when trials are many and the success chance is small.
With λ = 3, what is the chance of zero events?
e^(−3) ≈ 4.98%.
Is my data sent anywhere?
No. Everything is calculated in your browser, and nothing is stored or sent.