Poisson Distribution Calculator - Exact & Cumulative

About the Poisson Distribution Calculator

The Poisson distribution describes how many times an event happens in a fixed interval of time or space, when events occur independently at a steady average rate. How many calls will a help desk receive in the next hour? How many typing errors are on a page? How many customers will arrive in five minutes, or how many meteors will be seen in an hour of a meteor shower? If you know the average, the Poisson distribution tells you the chance of each possible count.

This Poisson distribution calculator takes the average rate, λ (lambda), and a number of events, k, and gives the probability of exactly k events, plus the cumulative probabilities of at most, fewer than, at least and more than k. It also shows the mean, variance and standard deviation, and the chance of no events at all.

How to Use the Poisson Distribution Calculator

Enter the average number of events per interval, λ. It does not need to be a whole number — 2.5 calls per hour is fine.

Enter the number of events, k, that you are interested in.

All five probabilities appear, with the mean and spread.

The Formula

  P(X = k) = e^(−λ) × λ^k ÷ k!

  mean = λ
  variance = λ
  standard deviation = √λ

Here e is the mathematical constant 2.71828… and k! is k factorial. Cumulative probabilities add the individual terms from 0 up to k.

Step-by-Step Example

A help desk receives 3 calls per hour on average. What is the chance of exactly 2 calls in the next hour?

  e^(−3) = 0.049787
  3^2 ÷ 2! = 9 ÷ 2 = 4.5
  P(X = 2) = 0.049787 × 4.5 = 0.2240 → 22.40%

  P(X ≤ 2) = e^(−3) × (1 + 3 + 4.5) = 0.049787 × 8.5 = 42.32%
  P(X ≥ 3) = 1 − 0.4232 = 57.68%

There is about a 22% chance of exactly two calls and a 58% chance of three or more.

A Second Example

Two emails arrive per hour on average. What is the chance of none in an hour?

  P(X = 0) = e^(−2) = 0.1353 → 13.53%
  P(X ≥ 1) = 86.47%

Roughly one hour in seven and a half will be quiet.

Scaling the Interval

Because events happen at a steady rate, λ scales with the interval. If a website gets 12 sign-ups per hour, it averages 1 per five minutes and 288 per day. To find the chance of no sign-ups in a five-minute window, use λ = 1: e^(−1) ≈ 36.8 percent. For a whole day, use λ = 288. Always make sure λ matches the interval in your question.

When the Poisson Model Applies

The Poisson distribution assumes that events are independent (one arrival does not make another more or less likely), that the rate is constant over the interval, and that two events cannot happen at exactly the same moment. Real data often departs from these assumptions. Shop customers arrive in bursts at lunchtime, so the rate is not constant; one car accident may cause another, so events are not independent. A useful check is to compare the mean and variance of your counts: in a Poisson process they are equal. If the variance is much larger, the data is overdispersed, and a negative binomial model may fit better.

Poisson and Binomial

The Poisson distribution is the limit of the binomial distribution when the number of trials is large and the probability on each is small, with λ = np. If 1,000 items each have a 0.2 percent chance of a defect, the number of defects is binomial with n = 1,000 and p = 0.002, but it is almost exactly Poisson with λ = 2. This makes the Poisson a convenient shortcut for rare events among many opportunities, such as mutations in DNA, insurance claims or misprints in a book.

Planning With Poisson Probabilities

Businesses use the Poisson distribution to plan capacity. If a small clinic sees 4 emergency walk-ins per day on average, how often will it see 6 or more? The calculator gives P(X ≥ 6) ≈ 21.49 percent, about one day in five. If the clinic can comfortably handle 7, the chance of exceeding that, P(X ≥ 8), is about 5.1 percent, roughly one day in twenty. Weighing such probabilities against staffing costs is the basis of queueing theory and call-centre planning.

Understanding Your Result

The headline is the probability of exactly k events.

The at most, fewer than, at least and more than lines give the cumulative probabilities.

The mean and spread line shows λ, its variance and standard deviation, and the chance of no events.

The worth knowing line restates the assumptions behind the model.

When Should You Use This Calculator?

Use it for arrivals, calls, emails, orders and other counts per interval.

Use it for defects, errors and other rare events among many opportunities.

Use it to plan staffing and capacity for unpredictable demand.

Use it for statistics homework on discrete distributions.

Common Mistakes

Using a λ that does not match the interval. Scale the rate to the time or space in the question.

Applying it when events cluster. Bursty or linked events break the independence assumption.

Confusing "at least" with "more than". At least 3 includes 3.

Expecting the variance to differ from the mean. In a true Poisson process they are equal.

Forgetting that k must be a whole number. Counts cannot be fractional, though λ can be.

Using the Poisson model for a fixed small number of trials. With a handful of trials and a sizeable chance of success, the binomial distribution is the right model.