How to use the poisson distribution calculator
- Enter the average number of events per interval, such as 3.5 calls per hour or 0.2 defects per meter. If you want to look at several intervals at once, set Number of intervals, which multiplies the average.
- Enter the count (k) you are asking about, a whole number from 0 up. Fill in the upper count only if you also want the chance of a range, from k up to that number.
- Read the exact probability first, then the cumulative ones: at most, fewer than, at least and more than k. The table shows the exact and at-most probabilities for counts from five below k to five above.
- Use Copy or Print to keep the figures. Change any number and the result updates at once.
Formula
P(X = k) = e^(-m) x m^k / k!
P(X <= k) = Q(k + 1, m)
P(X >= k) = P(k, m)
Mean = m, variance = m, standard deviation = sqrt(m)
m is the mean number of events in the period you are looking at (the average per interval times the number of intervals). P and Q are the regularized lower and upper incomplete gamma functions, which give the cumulative sums without adding up hundreds of terms.
Calls arriving at 3.5 per hour
With a mean of 3.5, the chance of exactly 2 calls in an hour is e^(-3.5) x 3.5^2 / 2 = 0.18496. The chance of 2 or fewer is 0.32085, and the chance of at least 2 is 0.86411. Because the exact value for 2 is counted in both, the two add to more than 1 by exactly 0.18496.
For a range, the chance of 2 to 5 calls is 0.72173. Over two hours the mean doubles to 7, so set Number of intervals to 2 and the distribution shifts and spreads, with a standard deviation of the square root of 7, about 2.6458.
When a Poisson model fits
The Poisson distribution describes a count of events in a fixed period or space when events happen one at a time, independently of each other, at a steady average rate. Typical examples are customer arrivals in an hour, typos on a page, or flaws along a length of cable.
Its most recognizable feature is that the variance equals the mean. If your real counts vary much more than their average, the events probably cluster, and a Poisson model will make rare large counts look less likely than they are.
Reading the cumulative probabilities
"At most k" adds the probabilities of 0 up to k. "At least k" adds k and everything above it. "Fewer than k" and "more than k" leave k itself out. The tool computes tail areas directly, so very small probabilities, such as 5.8553e-25 for 30 or more events when the mean is 2, keep their precision instead of being lost to rounding.
The values were compared with a plain sum of the exact probability mass function in a separate Python script. For counts from a binomial experiment with a fixed number of trials, use the binomial probability calculator instead.
Limits of this calculator
The mean after multiplying by the number of intervals can be up to 100,000, and the count up to 2,000,000. Probabilities are shown with five significant digits, and values smaller than 1e-300 appear as zero-like numbers in exponent form or as "less than 1e-300".
For related ideas, see the probability calculator and the z-score calculator, which handles the normal curve that a Poisson distribution resembles when its mean is large.
Assumptions and limits
- Events are assumed to occur independently and at a constant average rate during the whole period. Rush hours, bursts and learning effects break that assumption.
- The mean you enter is treated as exactly known. In practice it is an estimate from past data, so the true probabilities carry extra uncertainty.
- Counts must be whole numbers from 0 up. The mean must be above 0, and the upper count of a range must not be smaller than the count.
- The table shows counts from five below k (not below 0) to five above k, so it can be shorter than 11 rows when k is small.
Frequently asked questions
What is the difference between "at most" and "fewer than"?
At most k includes k itself, so it is the sum for 0 through k. Fewer than k stops one short, at k minus 1. For a count of 0, nothing is fewer than 0, so that probability is 0.
How do I use this for a different time period than my rate?
Multiply. If you get 3.5 calls an hour and want to look at a 30 minute window, enter 3.5 as the average and 0.5 as the number of intervals, which gives a mean of 1.75.
Why do the exact and cumulative probabilities add to more than 1?
They overlap on purpose. At most k and at least k both include the exact probability of k, so adding them counts that case twice. The pairs at most k with more than k, and fewer than k with at least k, each add to exactly 1.
Can I use it when the average is large?
Yes, up to a mean of 100,000. The probabilities are computed with logarithms and incomplete gamma functions so large means do not overflow, although for very large means a normal approximation would give similar numbers.