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✓ Editorially reviewed by Derek Giordano, Founder & Editor · BA Business Marketing

Poisson Distribution Calculator

Name: Poisson Distribution Calculator
Author: Derek Giordano

Exact and cumulative Poisson probabilities from an average rate, with the mean, variance and standard deviation

Updated October 2026

The Poisson distribution gives the probability of k events in an interval when events happen independently at a constant average rate μ: P(X = k) = μ^ke^−μ ÷ k! Its mean and variance both equal μ. In OpenStax’s bakery example, 12 loaves per half hour means μ = 2 loaves per five minutes, so the chance of exactly 3 in five minutes is 0.1804.

How Poisson Probabilities Are Calculated

First get μ for your interval: average count × (your interval ÷ the rate’s interval). Then P(X = k) = μ^ke^−μ ÷ k!, and the cumulative probability P(X ≤ k) adds the terms from 0 to k. Everything else follows from those: P(X < k) = P(X ≤ k) − P(X = k), P(X > k) = 1 − P(X ≤ k), and P(X ≥ k) = 1 − P(X < k).

Worked example (OpenStax Example 4.26): 12 loaves per 30 minutes is (5 ÷ 30) × 12 = 2 per five minutes. P(X = 3) = 2³ × e⁻² ÷ 3! = 8 × 0.135335 ÷ 6 = 0.1804.

Example	μ	Question	Probability
OpenStax 4.26: 12 loaves per 30 min, 5-minute window	2	P(X = 3)	0.1804
OpenStax 4.29: 0.75 calls per 15 min	0.75	P(X > 1)	0.1734
OpenStax 4.30: 147 emails	147	P(X ≤ 160)	0.8666
OpenStax 4.31: 1.7292 texts per hour	1.7292	P(X > 2)	0.2505
3 calls per hour	3	P(X = 5)	0.1008
Average 1.5 defects per batch	1.5	P(X ≤ 2)	0.8088
Rate scaled: 3 per hour, 30 minutes	1.5	P(X = 0)	0.2231

The OpenStax answers printed in the text match: 0.1734 (Example 4.29), 0.8666 (4.30) and 0.2505 (4.31). The last three rows are illustrations.

When Events Follow a Poisson Model

Use it for counts of events in a fixed interval of time, space or volume when the events happen one at a time, independently, at a steady average rate: calls to a help desk per hour, typos per page, defects per batch, arrivals per minute. It fails when events cluster (one breakdown triggering others), when the rate changes during the interval (lunch-hour rushes), or when there is a hard upper limit on the count, which is the binomial’s territory.

Mean, Variance and Rates Over Different Intervals

For a Poisson variable the mean and the variance are both μ, and the standard deviation is √μ; OpenStax’s 147 emails a day have a standard deviation of 12.1244. If the observed variance in your data is much larger than the mean, the counts are overdispersed and a Poisson model will understate the spread. Rates scale linearly with the interval: 3 calls an hour is 1.5 in 30 minutes and 72 in a day, provided the rate really is steady.

Poisson Approximation to the Binomial

When n is large and p is small, a binomial(n, p) count is close to Poisson with μ = np. OpenStax Example 4.32 uses n = 200 and p = 0.0102, so μ = 2.04; for 10 events the binomial gives 0.000039 and the Poisson 0.000045. The binomial probability calculator gives exact binomial results when you know n and p.

Limits of the Calculation

The calculator assumes the Poisson model fits; it does not test that. For very large μ the distribution is close to normal with mean μ and standard deviation √μ, and the normal distribution calculator can approximate it with a continuity correction.

Common Mistakes

Not rescaling the rate. A rate per hour must be converted before asking about 15 minutes.

Mixing up “at least” and “more than.” P(X ≥ 3) includes 3; P(X > 3) does not.

Using it for a fixed number of trials. Ten coin flips are binomial, not Poisson.

Assuming independence that is not there. Clustered events break the model.

Frequently Asked Questions

What is the Poisson distribution?

A probability distribution for the number of events in a fixed interval when events occur independently at a constant average rate μ. P(X = k) = μ^k e^(−μ) ÷ k!.

What does the Poisson distribution calculate?

The probability of a given number of events, or of at most or at least that many, in an interval, given the average number expected.

How do I use this Poisson calculator?

Enter the average count and the interval it covers, your interval length (the same if you are not rescaling) and k. The calculator returns P(X = k) and the cumulative probabilities.

How do you do a Poisson distribution on a calculator?

On TI calculators, poissonpdf(μ, k) gives P(X = k) and poissoncdf(μ, k) gives P(X ≤ k); this page returns the same values.

What are the mean and variance of a Poisson distribution?

Both equal μ; the standard deviation is √μ.

How does the Poisson distribution work?

It counts how many times an event happens in an interval when each moment is equally likely to bring an event and events do not affect one another, so only the average rate matters.

When can Poisson approximate the binomial?

When n is large and p is small; use μ = np. OpenStax’s example with n = 200 and p = 0.0102 gives nearly the same tiny probabilities.

How to Use This Calculator

Enter the average count — and the interval it covers.
Enter your interval — to rescale the rate (or the same length).
Enter k — the number of events.
Read the probabilities — exact and cumulative, with mean and SD.

🧮This calculator is part of our Math Calculators collection.

📚 Sources & References

[1] OpenStax. Illowsky B, Dean S. Introductory Statistics 2e, section 4.6 Poisson Distribution (Examples 4.26–4.32), 2023. Accessed Oct 22, 2026. openstax.org
[2] NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.19 Poisson Distribution. Accessed Oct 22, 2026. www.itl.nist.gov

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