Introduction
The Poisson Distribution Calculator finds the chance that a set number of events happen in a fixed time or space. It needs two inputs: the average rate (λ) and the number of events (x).
Use it for things that happen at a steady average rate, like calls to a help desk each hour, cars at a stop sign, or typos on a page. Pick the type of question you want to answer: exactly x events, fewer than x, at most x, more than x, at least x, or a range between two numbers.
You get the probability as a decimal and a percent, plus the mean, variance, and standard deviation. A step-by-step solution shows the Poisson formula with your numbers filled in, so you can check your work or study the method. A bar chart and a table show the full distribution, with your chosen values marked.
The second tab looks at the shape of the whole distribution for one λ value. It lists the mean, variance, standard deviation, mode, skewness, and excess kurtosis, and explains what each number means in plain words. You can save any result as a PDF.
How to use our Poisson Distribution Calculator
Type in your average rate and the number of events you want to test. You get back the probability, the percentage, the mean, the standard deviation, step-by-step math, a chart, and a full probability table.
Average Rate (λ): Enter the mean number of events in one time period or space. It must be more than 0. Example: 5 calls per hour.
Probability Type: Pick the question you want to answer. You can choose exactly x, fewer than x, at most x, more than x, at least x, or a range between two values.
Number of Events (x): Enter the whole number of events you want to check, like 2. Use 0 or any positive whole number. This box shows for all types except a range.
Minimum Events (x₁): When you pick the range option, enter the smallest number of events. This number is counted in the answer.
Maximum Events (x₂): Enter the largest number of events for the range. It must be the same as or bigger than x₁, and it is counted in the answer too.
Average Rate (λ) on the Distribution Properties tab: Enter one rate to see the mean, variance, standard deviation, mode, skewness, and excess kurtosis for that Poisson distribution.
Buttons: Click Calculate to see your results. Click Reset to go back to the default values, or Load Example to try a sample problem. Click Download Results to save or print your work.
What Is the Poisson Distribution?
The Poisson distribution tells you how likely it is that a certain number of events happen in a set amount of time or space. It works when events happen at a steady average rate, one at a time, and each event is independent of the others.
You only need one number to use it: λ (lambda). Lambda is the average number of events per interval. For example, if a shop gets 5 calls per hour on average, then λ = 5 for a one-hour interval.
The Poisson Formula
The chance of seeing exactly x events is:
P(X = x) = (e−λ × λx) ÷ x!
- λ = average number of events per interval
- x = the number of events you are asking about (0, 1, 2, 3, …)
- e ≈ 2.71828, a fixed math constant
- x! = x factorial, so 4! = 4 × 3 × 2 × 1 = 24
Mean, Variance, and Standard Deviation
In a Poisson distribution, the mean and the variance are both equal to λ.
- Mean (μ) = λ
- Variance (σ²) = λ
- Standard deviation (σ) = √λ
- Mode = the whole number just below or equal to λ
- Skewness = 1 ÷ √λ, so the shape gets more even as λ grows
When λ is small, the graph leans to the right. When λ is large (about 10 or more), the bars start to look like a normal bell curve.
When to Use It
Use the Poisson distribution for counts of rare or scattered events, such as:
- Calls arriving at a help desk each hour
- Cars passing a toll booth each minute
- Typos on a page of a book
- Flaws in a sheet of metal
- Emails you get each day
- Earthquakes in a region each year
Rules That Must Hold
- Events happen one at a time, not in pairs.
- The average rate stays the same over the whole interval.
- One event does not change the chance of the next event.
- x must be a whole number that is 0 or more, and λ must be greater than 0.
Quick Example
A bakery sells 4 birthday cakes per day on average, so λ = 4. The chance of selling exactly 2 cakes tomorrow is:
P(X = 2) = (e−4 × 4²) ÷ 2! = (0.018316 × 16) ÷ 2 = 0.1465, or about 14.65%.
Types of Poisson Probability
- P(X = x): exactly x events
- P(X < x): fewer than x events
- P(X ≤ x): at most x events (this is the cumulative probability)
- P(X > x): more than x events, found with 1 − P(X ≤ x)
- P(X ≥ x): at least x events, found with 1 − P(X < x)
- P(x₁ ≤ X ≤ x₂): between two counts, adding all terms in that range
Matching λ to Your Time Period
Lambda must match the interval you care about. If a store gets 12 customers per hour but you want a 15-minute window, use λ = 12 ÷ 4 = 3. If you want 3 hours, use λ = 12 × 3 = 36. Always scale the rate first, then compute the probability.
Poisson vs. Binomial
The binomial distribution counts successes in a fixed number of tries, like 10 coin flips. The Poisson distribution counts events with no fixed number of tries, just a rate over time or space. When a binomial has many trials (n large) and a small success rate (p small), the Poisson with λ = n × p gives a very close answer and is much easier to work out.