Poisson Distribution Calculator

What the Poisson distribution calculator does

The Poisson distribution describes how many times a random event happens in a fixed interval, when the events occur at a steady average rate. This calculator takes that rate and a number of occurrences and returns the chance of exactly that many, along with the cumulative chances above and below it.

It fits counts over time or space, from calls arriving at a switchboard to typos on a page. Below is how it works and the situations it suits.

How to use it

1. Enter the number of successes, the count you are asking about.
2. Enter the average rate, the mean number of events expected in the interval.
3. Press Calculate for the exact and cumulative probabilities, or Reset to clear it.

When a situation is Poisson

The Poisson distribution fits when events happen one at a time, independently, at a constant average rate over some interval, and you are counting how many occur. The interval can be a length of time, a stretch of road, an area, or a volume. What matters is that events land at random with no fixed number of trials behind them.

The number of customers arriving in an hour, the number of flaws in a metre of wire, the number of emails in a day: all are Poisson-like, because each is a count of independent events over an interval with a known average. This is what sets it apart from the binomial, which counts successes in a fixed number of separate trials. The Poisson has no set number of trials, only a rate.

How the probability is worked out

From the average rate, written as the Greek letter lambda, the chance of exactly x events is:

P(x) = λ^x × e^{(minus λ)} ÷ x!

Here e is the mathematical constant about 2.718. The formula balances two effects: the rate raised to the count grows with x, while dividing by x factorial shrinks quickly as x rises, and the exponential term scales everything so the probabilities across all counts add to 1. The result peaks around the average rate and tails off on either side.

The mean and variance are both the rate

The Poisson distribution has a striking property: its mean and its variance are equal, both equal to the rate lambda. The average number of events is the rate, which makes sense, and the spread of the counts happens to be exactly the same number.

This is a genuinely useful check. If you have count data whose average and variance are roughly equal, that is a hint the Poisson is a reasonable model for it. If the variance is far larger than the mean, the events are more clumped than Poisson allows, and a different model may fit better. So the equality of mean and variance is both a defining feature and a quick test of fit.

Its link to the binomial

The Poisson is closely related to the binomial distribution. When you have a very large number of trials, each with a very small chance of success, the binomial becomes awkward to compute but settles into the Poisson shape, with the rate lambda standing in for the number of trials times the tiny success chance.

This is why the Poisson is often called the distribution of rare events. It is the natural model when something could happen on any of a huge number of occasions but only rarely does, which is exactly the case for accidents, arrivals, and defects. In those settings the Poisson gives the same answer as the binomial would, with far less work.

A worked example

Suppose a call centre receives an average of 3 calls per hour, so the rate is 3, and you want the chance of exactly 5 calls in an hour.

The formula gives 3⁵ times e^-3, divided by 5 factorial, which is about 10.1 percent. Because the rate is 3, both the mean and the variance of this distribution are 3, so calls average three an hour with a spread that matches. The calculator also shows the chances of fewer than, at most, more than, and at least 5 calls, for the fuller picture.

Entering your values

Enter the count as a whole number and the average rate, which can be any positive value, not only a whole number. The probabilities are shown as decimals and percentages. The rate should describe the same interval you are asking about, so if you want the chance per hour, the rate must be the average per hour.

Questions people ask

What is the Poisson distribution?

The distribution of the number of independent events in a fixed interval when they occur at a constant average rate. It models counts over time, length, area, or volume.

How is it different from the binomial?

The binomial counts successes in a fixed number of separate trials. The Poisson counts events over an interval with no set number of trials, only an average rate.

Why are the mean and variance the same?

It is a property of the Poisson distribution: both equal the rate lambda. Roughly equal average and variance in count data is a sign the Poisson fits.

What is lambda?

The average rate, the mean number of events expected in the interval. It is the single number that defines a Poisson distribution.

References

A quick note on where the methods here come from. The Poisson distribution is set out in the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public statistics reference. OpenStax Introductory Statistics is a free, widely used textbook covering the Poisson distribution.

1. NIST/SEMATECH e-Handbook of Statistical Methods (the Poisson distribution). https://www.itl.nist.gov/div898/handbook/
2. OpenStax, Introductory Statistics (the Poisson distribution). https://openstax.org/details/books/introductory-statistics-2e