What this calculator does
The Poisson distribution gives the chance of a number of independent events in a fixed interval when only the average rate is known. With an average of 4 events, the chance of exactly 4 is 19.5% and the chance of 6 or fewer is 88.9%.
Its defining oddity is that the mean and the variance are the same number. A process averaging 4 events has a variance of 4 and therefore a standard deviation of 2, so the spread is fixed by the rate and there is no second parameter to adjust. If your data has a variance noticeably larger than its mean, it is overdispersed and Poisson is the wrong model.
The formula
The probability of exactly k events is e to the minus lambda, multiplied by lambda to the power k, divided by k factorial. The cumulative figure sums that from zero up to k. Lambda is the average number of events per interval, and the model assumes events are independent and occur at a constant average rate.
| Term | Meaning |
|---|---|
| λ (lambda) | The average number of events per interval. It is both the mean and the variance. |
| k | The number of events whose probability you want. |
| Independence | One event occurring must not change the chance of another. This is the assumption most often violated. |
| Overdispersion | Variance exceeding the mean, which means Poisson does not fit. |
The inputs explained
| Field | What to enter |
|---|---|
| Average rate (λ) | Average events per interval. It need not be a whole number. |
| Number of events (k) | The event count to evaluate. Must be a non-negative whole number. |
When to use it
Staffing for arrivals
Calls, customers or emergency admissions arriving at a known average rate, where the question is how often demand exceeds capacity.
Counting defects
Flaws per metre of material or per batch, where occurrences are rare and independent.
Modelling rare events
Equipment failures, accidents or decay counts, where the average is known but individual occurrences are not predictable.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What are the Poisson probabilities for an average of 4?
The chance of each event count for a process averaging four events.
| Events (k) | P(X = k) | P(X ≤ k) | P(X > k) |
|---|---|---|---|
| k = 0 | 1.83% | 1.83% | 98.2% |
| k = 2 | 14.7% | 23.8% | 76.2% |
| k = 4 | 19.5% | 62.9% | 37.1% |
| k = 6 | 10.4% | 88.9% | 11.1% |
| k = 8 | 2.98% | 97.9% | 2.14% |
Questions
When should I use the Poisson distribution?
When counting independent events in a fixed interval of time, space or volume, where you know the average rate and there is no fixed upper limit on how many could occur. If there is a fixed number of trials each with a success probability, the binomial distribution applies instead.
Why are the mean and variance the same?
It falls out of the mathematics of the distribution, which has only one parameter. The practical consequence is that Poisson cannot be fitted to a rate and a spread independently. If your data is more variable than its mean implies, a negative binomial model handles the extra dispersion.
What is the difference between Poisson and binomial?
Binomial has a fixed number of trials with a success probability each; Poisson has no upper limit and only a rate. Poisson is in fact the limit of the binomial as the number of trials grows and the probability shrinks with their product held constant, which is why it approximates rare binomial events well.
Can lambda be a decimal?
Yes. Lambda is an average, so 2.3 events per hour is perfectly ordinary even though you can never observe 2.3 events. Only k, the count whose probability you are calculating, has to be a whole number.
For a fixed number of trials instead, see the binomial distribution calculator. For the trial on which the first success occurs, see the geometric distribution calculator.