Poisson Distribution Calculator

Variance equals the mean, and why that is a test

Almost every distribution has separate parameters for location and spread. The Poisson has one number doing both jobs, and that is not a simplification — it is a constraint the data has to satisfy.

If the mean count is 4, the variance must be 4. You do not get to choose. So the ratio of observed variance to observed mean is a direct test of whether a Poisson can describe your data at all, and it takes one division.

Ratio near 1: a Poisson is plausible. The events are arriving independently at a steady rate, which is what the distribution assumes.

Ratio well above 1 — overdispersion. This is the usual finding in real count data. It means the events cluster: website traffic arrives in bursts when a link is shared, insurance claims cluster around storms, defects cluster when a machine drifts. The rate is not constant, or the events are not independent, or both.

Ratio well below 1 — underdispersion. Rarer, and it usually means the process is regulated rather than spontaneous: a scheduled service, a controlled feed, something with a mechanism preventing bunching.

The consequence of ignoring overdispersion is specific and always in the same direction: the model understates the tail. It will tell you that twenty arrivals in an hour is essentially impossible when your data has produced it three times this month. Since capacity planning and risk reserving are entirely about the tail, that is the part you were modelling for.

Not an approximation to the binomial — its limit

The Poisson is often introduced as “a good approximation to the binomial when n is large and p is small”. That is true and it undersells the relationship.

The Poisson is what the binomial converges to as n goes to infinity and p goes to zero with np held fixed at λ. The table in the tool shows it: with λ = 3 and k = 5, the binomial at n = 10 is 2.1 × 10⁻³ away, at n = 200 it is 2.5 × 10⁻⁴ away, and at n = 1000 it is 5.0 × 10⁻⁵ away. It walks in.

The reason this matters practically is that it collapses two parameters into one. A binomial needs n and p. A Poisson needs only their product.

And that is exactly right for the problems it is used on, because you frequently do not know n. How many people could have visited your site this hour? How many opportunities were there for a defect in this metre of cable? Those questions have no sensible answer. What you can observe is the average rate, and the Poisson needs nothing else.

The rule of thumb for using it as an approximation is n ≥ 20 with p ≤ 0.05, or n ≥ 100 with np ≤ 10. But the deeper point is that for arrival-type processes the Poisson is not standing in for a binomial you would rather have — it is the natural model.

What a Poisson process assumes

Three conditions produce a Poisson count. Each maps onto a way real data departs from it.

A constant average rate. Events arrive at the same expected rate throughout the interval. Web traffic breaks this immediately — the rate at 3am is not the rate at 9pm — and the fix is to model shorter intervals where the rate is roughly stable rather than to abandon the model.

Independence. One event does not make another more or less likely. This is what clustering breaks: one person sharing a link brings a hundred more, one machine fault causes a run of defects.

No simultaneity. Events do not arrive in exact coincidence; in a short enough interval you see at most one. Batch arrivals — a coach of tourists at a hotel desk — violate this, and produce overdispersion even when the average rate is perfectly stable.

All three are visible in the dispersion ratio, which is why one number is a reasonable first screen. Clustering, a varying rate and batch arrivals all inflate the variance above the mean.

A useful consequence when the conditions do hold: the time between events is exponential with mean 1/λ. Poisson counts and exponential gaps are two views of the same process, which is why queueing theory moves between them freely.

What to use when dispersion fails

Overdispersion is common enough that the alternatives are well developed. Which one depends on why the variance is inflated.

Negative binomial, the standard answer. It adds a second parameter that lets variance exceed the mean, and it has a natural interpretation: a Poisson whose rate itself varies from interval to interval. That is usually what is happening. It is the default for overdispersed counts in almost every statistical package.

Quasi-Poisson. Keeps the Poisson mean structure and simply inflates the standard errors by the estimated dispersion. Less principled, easy to fit, and often adequate when you only need honest uncertainty rather than good tail probabilities.

Zero-inflated models, when there are far more zeros than a Poisson allows. This happens when two processes are mixed — some units can never have an event at all. Counting cigarettes smoked in a population containing non-smokers is the standard example.

Shorter intervals, when the problem is a rate that varies over time rather than true clustering. Modelling each hour separately can restore the assumption without changing distribution.

The first move, though, is to look at why. Overdispersion is information: it says the events are not independent, and understanding the dependence usually matters more than patching the model.

The probability of nothing happening

P(X = 0) = e⁻ᵤ is worth knowing on its own, because it turns up in more places than the rest of the distribution.

At λ = 1 it is 0.368; at λ = 3, 0.050; at λ = 5, 0.0067. The decline is exponential, which means a modest rise in rate makes “nothing happened” very unlikely very quickly.

Reliability. The chance of no failures in a period is the survival probability, and e⁻ᵤ with λ = rate × time is the exponential reliability model.

Sampling and inspection. If you inspect and find nothing, e⁻ᵤ tells you what rates are compatible with that. Finding zero defects in 100 units does not mean the rate is zero — a rate of 3% would produce zero defects about 5% of the time.

That leads to the rule of three, which is genuinely useful: if you observe zero events in n trials, the upper 95% bound on the rate is about 3/n. Zero adverse reactions in 300 patients is consistent with a true rate up to about 1%. It follows directly from setting e⁻ⁿᴘ = 0.05, and it is the honest answer to “we saw none, so it is safe”.

How the shape changes with the rate

One parameter controls everything, and the tool redraws the distribution as you move it.

Small λ is strongly right-skewed. At λ = 0.5 the distribution is a spike at zero with a short tail: most intervals have no events, a few have one.

Large λ becomes symmetric and approaches a normal with mean and variance both λ. Past about λ = 20 a normal approximation is reasonable, though the exact calculation costs nothing here.

The skewness is 1/√λ — falling toward zero, never reaching it. A Poisson is always slightly right-skewed, because counts are bounded below by zero and unbounded above.

The most likely single count is ⌊λ⌋, and when λ is an integer both λ and λ−1 are jointly most likely. At λ = 3 the counts 2 and 3 are equally probable, which surprises people expecting a single peak at the mean.

A property that makes the distribution unusually convenient: Poissons add. The sum of independent Poisson counts with rates λ₁ and λ₂ is Poisson with rate λ₁+λ₂. So an hourly rate scales to a daily one by multiplying, and two independent sources combine by adding — no re-derivation needed.

Where it genuinely applies

Four settings, and the dispersion caveat for each.

Queueing and capacity. Arrivals at a till, a call centre, a server. The foundation of queueing theory. Watch for time-varying rates — model the peak hour rather than the day.

Reliability and defects. Failures per unit time, flaws per unit area. Usually well behaved when the process is stable, overdispersed when it is drifting — which makes the dispersion ratio a process-control signal in its own right.

Epidemiology. Rare disease counts in a region. Overdispersion here often indicates genuine clustering, which is frequently the finding rather than a nuisance.

Sport and other low-scoring counts. Goals in a football match are famously close to Poisson, which is why the model underpins so much match prediction. It is an approximation — goals are not quite independent, since a team that concedes changes how it plays — and it holds up better than it has any right to.

The common thread: the Poisson is the model for rare events in a continuum where you can observe a rate but not a number of trials. Where you can count the trials, use a binomial.

Sources and methodology

References for the distribution and for the dispersion diagnostics.

• NIST/SEMATECH e-Handbook — the Poisson distribution and its properties
• NIST Digital Library of Mathematical Functions §8.4 — the incomplete gamma identity the cumulative probabilities use
• Hanley & Lippman-Hand on the rule of three — what observing zero events actually bounds
• Binomial Distribution Calculator — the distribution this one is the limit of
• Normal Distribution Calculator — what a Poisson approaches at large λ
• How we calculate — methodology

Method. Probabilities are exact and computed in log space, so a large λ or a large k does not overflow through the factorial. Cumulative probabilities use the regularised incomplete gamma identity rather than summing terms, and the suite asserts that the closed form matches a literal term-by-term sum of the PMF across three rates, worst case under 1e-11. The binomial convergence table is computed live at four values of n rather than quoted, so the gaps shown are the actual ones. The dispersion ratio is a plain observed-variance over λ, deliberately left as a descriptive screen rather than dressed up as a formal test — a chi-square dispersion test exists, and the ratio is the part that changes a decision. That engine is verified on every change against 88 assertions, including that five discrete PMFs each sum to 1. The count and the per-case breakdown are published on the formula verification page.

Related calculators

Where this goes next:

More in Math, or browse all calculators.