What the Poisson Distribution Calculator does and when to use it
The Poisson distribution models how many times an event occurs in a fixed interval of time or space when events happen independently at a steady average rate. Typical cases include calls arriving at a help desk per hour, typos per page, defects per metre of cable, or radioactive decays per second. You enter the average rate, called lambda (λ), and the calculator returns the mean, variance, standard deviation, most likely count and a table of probabilities for each count.
The table lists P(X = k) and the cumulative probability for each number of events k, stopping once 99.9% of the probability is covered. To answer questions such as the chance of at least five events, subtract the cumulative value just below that count from 100%.
Formula and method
For a Poisson random variable X with average rate λ, the probability of exactly k events is P(X = k) = e^(−λ) × λ^k / k!. The mean and variance are both equal to λ, so the standard deviation is the square root of λ. The most likely value is the whole part of λ; when λ is a whole number, both λ − 1 and λ are equally likely.
The rate must match the interval. If calls arrive at 6 per hour and you care about 30 minutes, set λ = 3.
- λ the average number of events per interval; must be positive.
- k the count of events you are asking about (0, 1, 2, ...).
- e Euler's number, about 2.71828.
- k! k factorial, the product 1 × 2 × ... × k.
- Cumulative the running total P(X ≤ k).
Worked example
A shop receives an average of 3 online orders per hour. What does the distribution look like?
- Enter λ = 3. Mean = 3, variance = 3, and standard deviation = √3 = 1.7321.
- P(0) = e^(−3) = 0.049787, about 4.98%.
- P(3) = e^(−3) × 27 / 6 = 0.224042, so k = 2 and k = 3 are tied as the most likely counts.
- Cumulative through k = 4 is 81.53%, so the chance of five or more orders in an hour is 100% − 81.53%, about 18.5%.
The calculator prints exactly these figures and reports that 99.9% of outcomes fall at or below 10 events. For large rates above 200 the page omits the table and points to the normal approximation with the same mean and standard deviation.
Common mistakes and how to interpret the result
- Using the wrong interval. The rate and the question must refer to the same time or space. Scale λ when the window changes.
- Ignoring the assumptions. Events should be independent and occur at a constant average rate. Clusters, rush hours or contagion break the model.
- Expecting the variance to differ from the mean. If real data has a variance far above its mean (overdispersion), a Poisson model understates how often extreme counts occur.
- Reading the mode as the average. The most likely count can differ from λ because it must be a whole number.
Related calculators
- Binomial Distribution Calculator — fixed number of trials with a set success chance.
- Exponential Distribution — the waiting time between Poisson events.
- Normal Approximation Calculator — approximate large counts.
- Geometric Distribution — trials until the first success.