Poisson Distribution Calculator
Turn an average rate into Poisson probabilities: enter the rate per unit and the length of your interval, and get the chance of exactly k events, at most or at least k, or a range — with the mean, variance, a probability table and a bar chart. Free, no sign-up.
Leave at 1 if the rate already covers your whole interval.
Adds P(k ≤ X ≤ this value) to the results.
Results
λ = rate × length = 3 × 4 = 12
| Probability | Meaning | Value |
|---|---|---|
| P(X = 15) | exactly 15 | 0.0724 |
| P(X ≤ 15) | at most 15 | 0.8444 |
| P(X < 15) | fewer than 15 | 0.7720 |
| P(X ≥ 15) | at least 15 | 0.2280 |
| P(X > 15) | more than 15 | 0.1556 |
12
Mean (λ)
12
Variance (λ)
3.4641
Standard deviation (√λ)
λ = 12 is at least 10, so the normal approximation N(12, 3.4641²) is reasonable. With a continuity correction it gives P(X ≤ 15) ≈ 0.8438, against the exact 0.8444.
Probability distribution
| k | P(X = k) | P(X ≤ k) | P(X ≥ k) |
|---|---|---|---|
| 0 | 6.14 × 10−6 | 6.14 × 10−6 | 1.0000 |
| 1 | 7.37 × 10−5 | 7.99 × 10−5 | 1.0000 |
| 2 | 0.0004 | 0.0005 | 0.9999 |
| 3 | 0.0018 | 0.0023 | 0.9995 |
| 4 | 0.0053 | 0.0076 | 0.9977 |
| 5 | 0.0127 | 0.0203 | 0.9924 |
| 6 | 0.0255 | 0.0458 | 0.9797 |
| 7 | 0.0437 | 0.0895 | 0.9542 |
| 8 | 0.0655 | 0.1550 | 0.9105 |
| 9 | 0.0874 | 0.2424 | 0.8450 |
| 10 | 0.1048 | 0.3472 | 0.7576 |
| 11 | 0.1144 | 0.4616 | 0.6528 |
| 12 | 0.1144 | 0.5760 | 0.5384 |
| 13 | 0.1056 | 0.6815 | 0.4240 |
| 14 | 0.0905 | 0.7720 | 0.3185 |
| 15 | 0.0724 | 0.8444 | 0.2280 |
| 16 | 0.0543 | 0.8987 | 0.1556 |
| 17 | 0.0383 | 0.9370 | 0.1013 |
| 18 | 0.0255 | 0.9626 | 0.0630 |
| 19 | 0.0161 | 0.9787 | 0.0374 |
| 20 | 0.0097 | 0.9884 | 0.0213 |
| 21 | 0.0055 | 0.9939 | 0.0116 |
| 22 | 0.0030 | 0.9970 | 0.0061 |
| 23 | 0.0016 | 0.9985 | 0.0030 |
| 24 | 0.0008 | 0.9993 | 0.0015 |
| 25 | 0.0004 | 0.9997 | 0.0007 |
| 26 | 0.0002 | 0.9999 | 0.0003 |
| 27 | 7.75 × 10−5 | 0.9999 | 0.0001 |
| 28 | 3.32 × 10−5 | 1.0000 | 5.58 × 10−5 |
| 29 | 1.37 × 10−5 | 1.0000 | 2.26 × 10−5 |
| 30 | 5.50 × 10−6 | 1.0000 | 8.87 × 10−6 |
| 31 | 2.13 × 10−6 | 1.0000 | 3.37 × 10−6 |
| 32 | 7.98 × 10−7 | 1.0000 | 1.24 × 10−6 |
| 33 | 2.90 × 10−7 | 1.0000 | 4.45 × 10−7 |
| 34 | 1.02 × 10−7 | 1.0000 | 1.55 × 10−7 |
| 35 | 3.51 × 10−8 | 1.0000 | 5.23 × 10−8 |
How to Use the Poisson Distribution Calculator
- 1. Enter the average rate. The mean number of events per unit of time, distance, area or volume — 3 calls per hour, 0.8 typing errors per page.
- 2. Enter the length of your interval. Use the same units as the rate. The calculator multiplies the two to get λ, the expected count for your interval; leave the length at 1 if your rate already covers the whole interval.
- 3. Enter k, and optionally a second value. k is the number of events you are asking about. A second value adds the probability of the whole range between the two.
- 4. Read the results. Every probability for k, the mean, variance and standard deviation, a note on the normal approximation, and a probability table and bar chart.
What the Poisson Distribution Models
The Poisson distribution gives the probability of a number of events in a fixed interval when events happen independently, one at a time, at a constant average rate: calls to a help desk per hour, typing errors per page, radioactive decays per second, emergency admissions per night. Unlike the binomial there is no fixed number of trials, so the count can in principle be any whole number.
With λ the expected number of events in the interval, P(X = k) = e−λ λk / k!. The mean and the variance are both λ, so the standard deviation is √λ. The calculator computes cumulative probabilities through the regularised incomplete gamma function rather than by adding terms, so tail probabilities stay accurate.
Scaling the rate to your interval
λ always refers to the interval in the question. If a help desk averages 3 calls per hour, a 4-hour shift has λ = 3 × 4 = 12 and a 20-minute window has λ = 3 × 20/60 = 1. Rescale the rate, never the count: the probability of 12 calls in 4 hours is not the probability of 3 calls in one hour.
Worked Example: Calls to a Help Desk
Calls arrive at an average of 3 per hour. In a single hour (λ = 3) the chance of exactly 5 calls is e−3 × 35 / 5! = 0.1008, and the chance of a quiet hour with no calls at all is e−3 = 0.0498.
Over a 4-hour shift (λ = 12) the chance of exactly 15 calls is 0.0724, of 15 or more 0.2280, and of at most 15 0.8444. If the desk can handle 15 calls a shift, it will be overloaded on about 16% of shifts, since P(X > 15) = 0.1556.
Poisson or normal?
For λ of about 10 or more the Poisson is close to symmetric and resembles a normal distribution with mean λ and standard deviation √λ. For the 4-hour shift the continuity-corrected approximation gives P(X ≤ 15) ≈ Φ((15.5 − 12) / 3.4641) = 0.8438, against the exact 0.8444. For small λ the distribution is strongly right-skewed and the approximation fails, so use the exact values.
Common mistakes
- Forgetting to rescale λ. Applying a rate per hour to a 4-hour question underestimates every expected count fourfold.
- Using the Poisson for clustered events. Accidents after a storm or infections within a household are not independent; the variance then exceeds the mean and the Poisson understates the tails.
- Reading P(X ≥ k) as 1 − P(X ≤ k). It is 1 − P(X ≤ k − 1): the probability of exactly k belongs to both "at least k" and "at most k".
Frequently Asked Questions
- What is the Poisson distribution used for?
- It models counts of independent events in a fixed interval of time or space at a constant average rate — calls per hour, defects per metre of cable, goals per match, mutations per genome. It also approximates the binomial well when the number of trials is large and the probability of success small.
- How do I calculate a Poisson probability?
- Find λ, the expected number of events in your interval, then use P(X = k) = e−λ λk / k!. For λ = 3, P(X = 5) = e−3 × 243 / 120 = 0.1008. Cumulative probabilities add these terms, and the calculator evaluates them all exactly.
- Why are the mean and variance of a Poisson distribution equal?
- Both equal λ, a consequence of events arriving independently at a constant rate. Comparing a sample's variance with its mean is therefore a quick check on the model: a variance much larger than the mean, called overdispersion, points to clustering, and a negative binomial model may fit better.
- When can I use the Poisson approximation to the binomial?
- When n is large and p small — common guides are n ≥ 20 with p ≤ 0.05, or n ≥ 100 with np ≤ 10. Set λ = np. For example, 1,000 items with a 0.2% defect rate give λ = 2, and the chance of no defects is e−2 = 0.1353.
- What does λ mean in the Poisson distribution?
- λ (lambda) is the expected number of events in the interval you are asking about: the average rate multiplied by the length of the interval. It is both the mean and the variance of the count, and it must match the interval — 3 per hour becomes 12 per 4 hours.
- How large can λ be?
- Up to 10,000. Beyond that the series behind the cumulative probabilities starts to lose accuracy, and a normal distribution with mean λ and standard deviation √λ is an excellent approximation anyway. The table and chart are drawn while they fit in about 200 rows, which means λ up to roughly 250.