Poisson Distribution Calculator

How to use

1. 1Pick a region: P(X = k) for an exact count, P(X <= k) or P(X < k) for a left tail, P(X >= k) or P(X > k) for a right tail, or P(a <= X <= b) for a range.
2. 2Enter lambda, the mean number of events per interval. Any non-negative number works (decimals welcome), up to 1,000,000.
3. 3Enter k for a single-bound region, or both a and b for the range region. All counts must be non-negative whole numbers.
4. 4Read the probability as a decimal and a percent, plus the mean, variance, standard deviation, mode, skewness, and excess kurtosis. The PMF bar chart shades the bars inside your chosen region.
5. 5Use the example chips to load classic textbook and applied scenarios in one click, or Copy summary or Copy full report to paste the numbers into homework, a spreadsheet, or a launch doc.

About this tool

Poisson Distribution Calculator returns the probability of seeing a given number of events in a fixed interval, given the mean rate lambda. It covers every probability region a student, A/B testing analyst, queue or reliability engineer, epidemiologist, or sports model builder reaches for next to the answer. Pick one of six regions in a single click: P(X = k) for an exact count (the textbook poissonpdf), P(X <= k) and P(X < k) for left-tail cumulative probabilities (poissoncdf), P(X >= k) and P(X > k) for right-tail probabilities, and P(a <= X <= b) for a range that is closed on both ends. Enter lambda as a non-negative number (any rate up to 1,000,000) and the result panel updates instantly with the probability to six significant figures and to four decimal places of percent, the mean lambda, the variance lambda, the standard deviation sqrt(lambda), the mode floor(lambda), the skewness 1/sqrt(lambda), and the excess kurtosis 1/lambda. A live SVG bar chart of the PMF shades the bars inside the requested region in blue and the rest in slate, so the answer is visible at a glance; for large lambda the chart automatically zooms to the most probable bars around the mode so the visualization stays readable. Numerically, the PMF is evaluated in log space using a Lanczos log-gamma approximation for log(k!), and the CDF is summed outward from the mode so the dominant terms anchor the running total. This stays stable and precise for lambda up to a million with sub-millisecond evaluation in modern browsers, which the naive lambda^k e^(-lambda) / k! formula cannot do (it overflows past k around 170). Useful for high-school and college statistics homework, AP Statistics poissonpdf and poissoncdf questions, queueing theory (calls per hour, customer arrivals per minute, packets per second), defect-rate and reliability work in quality control, accident-rate and insurance modeling, web traffic spike modeling, sports scoring models (goals per match), email spam volume modeling, radioactive decay counts, and any scenario where independent events arrive at a steady average rate. Example chips load common textbook and real-world scenarios (calls per hour, defects per batch, visitors per minute, accidents per month, emails per hour, goals per soccer match) so you can verify a class problem or a back-of-envelope estimate in one click. Everything runs locally in your browser, so the numbers you type for class, work, or research never leave your device. Pair this tool with the Binomial Distribution Calculator when you have a fixed number of trials, with the Normal Distribution Calculator when lambda is large and you want a continuous approximation, or with the P-Value Calculator when you have already turned the count into a test statistic.

Free to use. Works in your browser. No signup, no login.