POISSON in Calligra Sheets: Model a Count in a Fixed Interval

=POISSON(count; mean; cumulative)

All three arguments are required. Count should be a nonnegative integer. Mean is the expected number of events in the same fixed interval, not a rate expressed in a different interval. Use 0 for exact-count probability and 1 for the cumulative probability; the source branches on zero versus nonzero.

For positive mean lambda, the point probability is exp(−lambda)×lambdaˣ/x!. The cumulative probability sums masses from zero through the count. A mean need not be an integer: an average of 2.5 calls per interval is meaningful even though each observed count is whole.

Assume a process has a mean of 3 arrivals per ten-minute window and is suitably modeled by independent constant-rate arrivals.

1. Enter 2 in A2 labeled Arrival count.
2. Enter 3 in B2 labeled Mean per 10 minutes.
3. Enter =POISSON(A2;B2;0) in C2. Expected exact-count probability: 0.224041807655.
4. Enter =POISSON(A2;B2;1) in D2. Expected at-most-two probability: 0.423190081127.
5. Enter =1-D2 in E2. Expected more-than-two probability: 0.576809918873, or 57.68%.

The cumulative event includes zero, one and two arrivals. It is not the probability of exactly two. Check the interval definition before applying the result to capacity or scheduling.

If the same constant-rate process is observed for twenty minutes, the modeled mean becomes 6, not 3. An expression such as =POISSON(2;6;1) then answers the probability of at most two arrivals in the longer window. This rate-times-duration rescaling assumes the rate stays constant; it is not valid merely because the window doubled.

For mean 3, the no-arrival probability is =POISSON(0;3;0), expected 0.0497870683679, about 4.98%. A Poisson model has variance equal to its mean; clustered, dependent or systematically varying arrivals can violate its assumptions.

The inspected implementation handles mean = 0 with special branches: noncumulative mode returns 0, while cumulative mode returns 1 for admitted nonnegative counts. Consequently, POISSON(0;0;0) is source-predicted to return 0.

Mathematically, a zero-mean Poisson variable is always zero, so P(X=0) should be 1. The returned zero is an implementation discrepancy, not the correct degenerate-distribution probability. Avoid relying on this edge case: handle a zero-mean model explicitly, or use a separately validated implementation. Do not infer from the cumulative result 1 that the point-mass branch is also correct.

For x = 2.5 and mean 3, the source’s point calculation raises 3 to the original 2.5 but applies factorial after integer conversion, producing approximately 0.388051793879. In cumulative mode it loops only through the converted integer 2, yielding approximately 0.423190081127.

There is no event “exactly 2.5 arrivals” in the discrete count model. These outputs are not a supported continuous extension, and the two modes do not apply the same fractional-input treatment. Validate count as an integer before calling POISSON.

Negative count or mean constructs an internal value error. Zero mean is accepted despite the handbook’s positive-parameter wording, with the defect above. Large counts and means can stress the direct power/factorial/summation implementation; no general high-tail accuracy guarantee is asserted. Cumulative values very close to one can make a complement lose precision.

KDE EXPONDIST models a waiting time under a related constant-rate process; KDE NEGBINOMDIST uses a success-target trial model. Preserve both the meaning of the mean and the cumulative flag when migrating a workbook, and retest zero/fractional boundaries rather than assuming cross-product parity.

Checked October 18, 2026. Original introduction date is unknown. POISSON is registered in both inspected source versions, Calligra 25.04.2 and 26.08.2. Their relevant statistical-function, calculation-engine and converter source files are byte-identical. Version inspection establishes availability in those releases, not a first-introduction date or a runtime guarantee for every platform.

Primary sources: KDE POISSON handbook, 26.08.2 implementation, 25.04.2 statistical functions, and numeric converter. Mathematical values above are independent calculations. Native observations below were obtained in Calligra Sheets 25.04.2 on Linux, October 18, 2026; no native 26.08.2 execution is claimed. The handbook is a living reference and can differ from the pinned implementation at the explicitly described boundaries.

The following formula results were obtained in the actual application and checked in its completed native-saved workbook. The screenshot is an unmodified application capture. The values in this table are native results; the earlier reference values and model calculations remain separately identified. Small final-digit differences do not make the independent reference a native observation.

The native zero-mean point result 0 below is the implementation discrepancy described above, not the mathematically correct mass at zero. Fractional-count outputs are recorded to expose the input-handling inconsistency, not recommended as valid count probabilities.

The completed saved values, including error text, determine this table; display rounding can show fewer digits. These tests do not establish all locales, input types, extreme numerical ranges or cross-application import behavior.