Poisson Distribution Calculator — probabilities for event counts

The Poisson distribution calculator above computes the exact probability of observing k events when events occur independently at a known average rate. It gives you the point probability P(X = k), all four cumulative tails, the distribution's mean, variance and standard deviation, and a probability bar for the counts nearest yours. A separate interval field lets you scale the rate, so three calls per hour over four hours becomes a lambda of twelve without you doing the multiplication.

Arb Digital keeps this tool alongside the rest of the statistics set because Poisson is the distribution behind most real counting problems — support tickets per day, defects per production run, conversions on a low-volume campaign — and it is routinely misapplied to data that violates its central assumption. The sections below spend as much space on when Poisson is wrong as on how to compute it, because that is where the expensive mistakes live.

What This Poisson Distribution Calculator Does

A Poisson random variable counts how many times something happens in a fixed window, when the events are independent and arrive at a constant average rate. Crucially there is no fixed number of trials. You are not asking "how many of these 50 attempts succeeded" — that is a binomial question. You are asking "how many events landed in this hour", where the answer could in principle be any non-negative whole number.

The calculator evaluates the probability mass function directly, then sums it to build cumulative values. It reports P(X = k), P(X < k), P(X ≤ k), P(X ≥ k) and P(X > k) together, along with the mode — the single most likely count — because the most likely outcome is frequently not the one people expect when lambda is small. Every probability is computed through the log-gamma function so that large values of k or lambda stay accurate rather than overflowing.

How to Use It

1. Enter your average rate. This is the mean number of events in one interval, taken from historical data. It does not have to be a whole number — 2.7 tickets per hour is a perfectly valid rate.
2. Set the number of intervals. Leave it at 1 to work in the units you entered, or raise it to scale up. The effective lambda is shown beneath the buttons.
3. Enter k, the event count you want the probability of. Unlike the binomial there is no maximum.
4. Read the correct tail. "At least k" includes k; "more than k" does not. Both appear in the panel so the choice is explicit rather than assumed.
5. Look at the bars. They show the shape of the distribution around your k, which makes it obvious whether your value sits near the peak or far out in the tail.

The Formula: How Poisson Probability Is Calculated

The Poisson probability mass function is P(X = k) = (λ^k × e^−λ) ÷ k!, where λ is the expected number of events in the interval and e is roughly 2.71828. The term e^−λ is the probability of zero events, λ^k scales for the k occurrences, and dividing by k! removes the ordering, since the events are indistinguishable and the order they arrived in does not matter.

The most striking property of the distribution is that the mean and the variance are both λ. One parameter fixes both the centre and the spread, which is unusual and enormously useful as a diagnostic. The standard deviation is √λ, which means that as the rate grows the absolute spread grows but the relative spread shrinks: at λ = 4 the standard deviation is 2, or 50% of the mean; at λ = 100 it is 10, or 10% of the mean. That is why low-volume counts look so erratic and high-volume ones look stable, without anything having actually changed.

Cumulative probabilities are sums of the point values. P(X ≤ k) adds the terms from 0 to k, and P(X ≥ k) is 1 − P(X ≤ k − 1). Because the distribution has no upper bound, "at most k" is always a finite sum while "at least k" is computed as a complement rather than an infinite one. This calculator evaluates each term as exp(−λ + k ln λ − ln k!), using the log-gamma function for ln k!, so that large counts do not overflow the way a direct factorial would.

A Worked Example You Can Check Yourself

The calculator loads with λ = 2 and k = 3, the standard textbook case. Work it by hand: λ^k = 2³ = 8, e⁻² = 0.1353353, and k! = 3! = 6. So P(X = 3) = (8 × 0.1353353) ÷ 6 = 1.0826822 ÷ 6 = 0.180447. That is what the tool prints.

The cumulative value is just as checkable. P(X = 0) = e⁻² = 0.135335. P(X = 1) = 2 × e⁻² = 0.270671. P(X = 2) = (4 × e⁻²) ÷ 2 = 0.270671 — identical to P(X = 1), which is a real feature of Poisson when λ is a whole number, not a rounding coincidence. Adding those three to P(X = 3) gives P(X ≤ 3) = 0.857123, so P(X > 3) = 0.142877 and P(X ≥ 3) = 1 − P(X ≤ 2) = 0.323324. The mean and variance are both 2, and the standard deviation is √2 = 1.4142.

A second case with a non-integer rate, since real rates rarely land on whole numbers: λ = 3.6 and k = 2 gives P(X = 2) = 0.177058. The NIST/SEMATECH e-Handbook of Statistical Methods is the reference to check the definitions against.

The Equal Mean and Variance Test Nobody Runs

Poisson makes an unusually strong claim: the variance equals the mean. That gives you a free diagnostic almost nobody uses. Take your historical counts, compute their mean and their variance, and compare. If the variance is close to the mean, a Poisson model is plausible. If the variance is much larger, your data is overdispersed and Poisson will badly understate the chance of extreme counts.

Overdispersion is the norm rather than the exception in business data, and the cause is almost always that the rate is not constant. Support tickets arrive faster on Mondays. Website conversions spike during a promotion. Machine defects cluster when a tool wears down. Every one of those means the single λ you fitted is an average across genuinely different regimes, and averaging rates does not average their variability — it hides it. A staffing plan built on a Poisson model fitted to overdispersed ticket data will be understaffed on exactly the days it matters. You can run that check with our variance calculator and the mean, median and mode calculator in under a minute, and it is worth doing before trusting any Poisson figure.

Underdispersion, where variance is well below the mean, is rarer and usually means the events are not independent in the opposite direction — something is regularising them, like a scheduled process or a rate limit.

When the Poisson Model Is the Wrong One

Four conditions define a Poisson process, and each fails in a recognisable way.

Events must be independent. One occurrence must not make another more or less likely. Failures that cascade — one server going down and taking others with it — cluster badly and produce far more extreme counts than Poisson predicts.

The rate must be constant across the interval. If you fit a single rate to a day that includes both a quiet night and a busy afternoon, the model describes neither. Split the interval instead of averaging across it.

Two events cannot occur at exactly the same instant. In practice this fails when events arrive in batches. A single broken deployment generating forty error reports at once is one event, not forty, and counting them as forty inflates both the rate and the apparent randomness.

There must be no fixed ceiling. If you are counting successes out of a fixed number of attempts, the count is bounded by that number, and the binomial distribution calculator is the correct tool. Poisson allows any count, which is fine for tickets per day but wrong for conversions among exactly 200 visitors.

Poisson, Binomial and Normal: Which One Applies

The three distributions are connected, and knowing the connections tells you which to reach for. Binomial counts successes in a fixed number of trials with a known success chance. When the number of trials is large and the per-trial chance is small, the binomial converges on Poisson with λ = np — which is why Poisson is often called the law of rare events. The practical consequence is that a very low click-through rate over a very large number of impressions can be modelled either way with nearly identical answers.

Going the other direction, when λ itself becomes large — above roughly 20 or 30 — the Poisson distribution becomes approximately normal with mean λ and standard deviation √λ. At that point a continuous model is easier to work with, and our normal distribution calculator takes over, though a half-unit continuity correction at the boundary improves the match noticeably. Poisson sits between the two: unbounded like the normal, discrete like the binomial. If you need the chance of a simple compound event rather than a count, the probability calculator is the simpler route.

Reading a Small Lambda Correctly

When λ is below 1, the distribution is heavily skewed and the single most likely outcome is zero. At λ = 0.5, P(X = 0) is 0.607 — the most probable result is that nothing happens at all, even though the average is half an event. Missing that leads to a persistent misreading of low-volume data, where a run of zeros gets interpreted as a change when it is simply the most likely thing the model predicts.

The mode of a Poisson distribution is the floor of λ, and when λ is a whole number there are two equally likely modes: λ and λ − 1. That is why P(X = 1) and P(X = 2) came out identical in the worked example above. The calculator prints the mode alongside the tails so this is visible rather than surprising. When you are comparing observed counts against expected ones across several categories rather than testing a single count, the chi-square calculator is the appropriate test, and the Penn State Eberly College of Science statistics courses cover the goodness-of-fit procedure for Poisson data in detail.

Common Mistakes to Avoid

• Using Poisson when there is a fixed number of trials — a count that cannot exceed a known ceiling is binomial, and Poisson will assign real probability to impossible outcomes above it.
• Fitting one rate across intervals that clearly differ — averaging a busy period and a quiet one produces a model that understates the variability of both.
• Ignoring overdispersion — if the observed variance is well above the mean, the Poisson tail probabilities are too small and extreme counts will happen more often than predicted.
• Forgetting to scale lambda to the interval — a rate of 3 per hour over 8 hours is a lambda of 24, not 3. The interval field exists to stop this.
• Counting batched events individually — forty alerts from one incident is a single event for modelling purposes, and treating it as forty corrupts both the rate and the spread.

Related Free Tools From Arb Digital

For counts out of a fixed number of trials, use the binomial distribution calculator. For continuous measurements, the normal distribution calculator applies instead. Test observed against expected counts with the chi-square calculator, check whether your data is overdispersed with the variance calculator, and work out simple event odds with the probability calculator. The free online tools hub has the full statistics set.

Frequently Asked Questions

This calculator performs a probability computation on the values you enter. Whether a Poisson model fits your data depends on whether the rate is genuinely constant and the events genuinely independent, which the tool cannot verify.