The Poisson distribution describes how many times an event happens in a fixed interval when events occur independently at a steady average rate. It is the go-to model for arrivals, defects, accidents and other counts with no fixed upper limit. Give the calculator the average count λ (or a rate and an interval length) and the number of events you are interested in, and it returns exact and cumulative probabilities, the formula with your numbers, a highlighted chart and a probability table.
How to use the Poisson distribution calculator
- Enter λ, the mean number of events per interval, or switch to rate × length and enter, for example, 2.5 defects per square meter and 2 square meters.
- Choose the event: exactly k, at most, fewer than, at least, more than, or between k and k₂.
- Enter k (and k₂ for a range).
- Read the probability on the tape and see which bars it covers in the chart. The table lists P(X = k), P(X ≤ k) and P(X ≥ k) across the likely range.
Poisson formula
for k = 0, 1, 2, … The mean and variance are both λ, and the standard deviation is √λ. Cumulative probabilities add the terms; internally the calculator evaluates them with the regularized incomplete gamma function, which stays accurate even for λ in the millions:
Worked example
A small IT help desk receives an average of 4 calls per hour.
- Exactly 2 calls next hour: e−4 × 4² ÷ 2! = 0.018316 × 16 ÷ 2 = 0.146525, about 14.7%.
- No calls at all: e−4 = 0.018316, under 2%.
- At least 7 calls, enough to overwhelm one technician: 1 − P(X ≤ 6) = 1 − 0.889326 = 0.110674. In roughly 1 hour in 9, the desk will need backup.
- The standard deviation is √4 = 2 calls, so most hours see between 2 and 6 calls.
Using a rate. Fabric has 2.5 flaws per square meter on average. A 2 m² panel therefore has λ = 2.5 × 2 = 5. The chance that a panel has at most 3 flaws is 0.006738 + 0.033690 + 0.084224 + 0.140374 = 0.265026.
Applying the Poisson model well
Check the assumptions
- Constant rate. Calls per hour at 3 a.m. and at noon follow different rates; model each period separately.
- Independence. One event must not trigger another. Aftershocks following an earthquake violate this.
- No simultaneous events. Events should arrive one at a time.
Mean versus variance
With historical counts, compare the sample mean and variance. If the variance is roughly equal to the mean, the Poisson model is reasonable. A variance two or three times the mean signals clustering; a negative binomial model fits such data better. A chi-square goodness-of-fit test can formally compare observed counts with Poisson expectations, remembering to subtract a degree of freedom for estimating λ.
Large λ
As λ grows, the Poisson distribution looks more and more like a normal curve with mean λ and SD √λ. For λ of 20 or more the calculator also prints the normal approximation with a continuity correction so you can see how close it is. For counts out of a fixed number of trials, use the binomial probability calculator instead.
Frequently asked questions
What kinds of data follow a Poisson distribution?
Counts of independent events over a fixed stretch of time, space or volume, when events occur at a steady average rate and two cannot happen at exactly the same instant. Typical examples are calls per hour, typos per page, flaws per square meter of fabric and website errors per day.
What does λ (lambda) mean?
λ is the average number of events in the interval you are asking about. If a help desk averages 12 calls per hour and you care about a 20-minute window, λ = 12 × (20 ÷ 60) = 4. Use the rate × length option to have the calculator do that multiplication.
Why are the mean and variance both λ?
It is a defining property of the Poisson distribution. A quick check on real data: if the sample variance of your counts is much larger than the mean, the events are clumped (overdispersed) and a Poisson model will understate the chance of extreme counts.
How is the Poisson related to the binomial?
When the number of trials n is large and the success probability p is small, Binomial(n, p) is very close to Poisson(λ = np). For example, the number of winners among 10,000 tickets each with a 1-in-5,000 chance is close to Poisson with λ = 2.
Can λ be a decimal?
Yes. λ is an average, so 2.7 or 0.35 are fine. Only the count k must be a whole number, because you cannot observe 2.5 events.