Poisson Distribution Calculator

Enter an average event count and observed count to calculate exact and cumulative probabilities plus distribution summaries.

Average event count (λ)

Observed events (k)

—P(X = k)

—P(X ≤ k)

—P(X ≥ k)

—Standard deviation

—For a Poisson distribution, mean and variance both equal λ.

Core describe → test → interpret workflows

Use descriptive context first, choose the method that matches the design, then keep inputs, settings and result evidence reproducible instead of copying a p-value without its analysis recipe.

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Poisson range probability & distribution evidence

The exact, at-most, and at-least probabilities above remain first. This optional workspace provides inclusive between/outside evidence and a probability table generated by the same WTAStats Poisson PMF/CDF engine.

Inclusive lower countInclusive upper count

k	P(X=k)	P(X≤k)	Relative mass

Probability evidence

Range endpoints are inclusive.

Interpret statistical results in context

Use plots and descriptive summaries before formal tests. Check the sampling/design assumptions that matter for the chosen method, report effect size and uncertainty where available, and avoid treating a threshold such as p < .05 as proof of importance, causation, or truth. Browser calculations are educational/planning utilities, not domain-specific professional advice.

Practical guide and verification

Use the tool first, then apply these checks to verify the inputs, interpret the result, and hand it off without displacing the primary workflow.

Match lambda to the same exposure interval as the question

A rate of four arrivals per hour cannot be used directly for a ten-minute interval. Scale the average count to the interval, area, distance, or exposure being modeled before reading exact, cumulative, or range probabilities.

Choose inclusive boundaries explicitly

Probability wording such as at most, fewer than, at least, more than, and between maps to different integer boundaries. The range workspace states its endpoints as inclusive so you can translate the question before calculating instead of correcting an off-by-one error afterward.

Check whether a Poisson model is plausible

The model assumes count events occurring with a stable average rate and an independence structure appropriate to the application. Clustering, capacity limits, changing rates, or strong dependence can make a numerically correct Poisson probability a poor description of the real process.

Use the displayed distribution mass as a numerical check

The probability table shows how much mass is covered by the displayed k range and how much remains in the upper tail. A range result should agree with the highlighted probability rows and with the complement shown beside it.