Poisson Distribution Calculator
Calculate Poisson probabilities for counts of events in a fixed interval, with cumulative values, the distribution table and a check of whether the constant rate assumption is plausible.
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Counting events in a fixed interval
The Poisson distribution describes how many times something happens in a fixed window when the events occur independently at a constant average rate. Support tickets per hour, defects per square metre, arrivals at a queue, radioactive decays per second and server errors per minute all fit it when the events do not influence each other. Its distinguishing feature is that the variance equals the mean, so a rate of nine events per hour implies a standard deviation of three, and seeing anywhere from three to fifteen in an hour is entirely ordinary.
The constant rate assumption is the one that fails
Real arrival rates are rarely constant. Support tickets peak during business hours, traffic peaks in the evening, and errors cluster around a deployment. When the rate varies, the observed variance exceeds the mean, which is called overdispersion and is the standard sign that a Poisson model does not fit. Events that trigger each other, such as a failure causing a retry storm, break the independence assumption in the same direction.
Relationship to the binomial
Poisson is the limiting case of the binomial when the number of trials is large and the probability of each is small, with their product held fixed. This is why it is the right model for rare events in large populations, and why using the normal approximation to the binomial fails exactly where Poisson works. For a rate below about ten the distribution is visibly skewed, and only for larger rates does it start to look symmetric.
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Frequently Asked Questions
What does it mean that the variance equals the mean?
That the spread is fixed by the rate rather than being a free parameter. A rate of nine implies a standard deviation of three, so counts from about three to fifteen are unremarkable. If your data spreads more widely than that, the Poisson model does not fit.
What is overdispersion?
Observed variance larger than the mean, which is the standard symptom of a rate that is not constant or events that are not independent. Peak hours, deployments and retry storms all produce it, and a negative binomial model is the usual replacement.
How is Poisson related to the binomial?
It is the limit of the binomial when the number of trials grows large and the individual probability shrinks, with their product fixed. That is why it fits rare events in large populations, exactly the regime where the normal approximation to the binomial fails.
Can the rate be a decimal?
Yes. The rate is an average over the interval and need not be a whole number, even though the count itself must be. A rate of 0.4 failures per day is perfectly meaningful and gives a high probability of zero failures on any given day.
How do I change the interval?
Multiply the rate by the ratio of the intervals. A rate of 3 per hour is 72 per day and 0.05 per minute. The distribution shape changes with the rate, so probabilities do not scale in the same simple way that the rate does.
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How to Use
Enter the average rate and a count to get the probability of that many events.
Disclaimer: This tool is provided "as is" without warranty of any kind. Results are for educational and utility purposes.
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