POISSON in Zoho Sheet: Count Probabilities and Cumulative Events

Use POISSON when you want the probability of a count of occurrences in a fixed time, area or other exposure under a Poisson statistical model. The official Zoho Sheet reference documents POISSON(x; λ; mode). The x argument is a count of events, λ (lambda) is the expected mean count for the same exposure, and the third argument switches between exact-count probability (0 or FALSE) and cumulative probability at or below the count (1 or TRUE). These are different questions. Z1

For example, =POISSON(2;3;FALSE) is mathematically expected to return 0.224041807655 under a Poisson mean of three events per period: about a 22.40% chance of exactly two events. =POISSON(2;3;TRUE) is independently expected to return 0.423190081127: about a 42.32% chance of zero, one or two events. Both numbers are independent SciPy/model calculations, not Zoho-native outputs.

Evidence scope: Zoho’s English public Help supports the function name, argument order, mode meanings and its named errors. No formula was executed in Zoho Sheet, and the guide does not assert mobile/desktop parity, specific minimum versions or undocumented coercion behaviour.

POISSON(x; λ; mode)

Zoho’s help explicitly lists #NUM! when x<0 or λ<0 and describes both Boolean and numeric mode values. It does not specify how noninteger x is handled (reject, truncate, round or something else) or a default when mode is omitted. Statistical Poisson theory defines discrete event counts; use integer inputs and test any rounding in advance in the target app instead of importing Excel’s documented truncation behaviour. The function’s λ is a count mean, not the gamma-distribution rate or scale. Z1

Mode wording: Zoho calls the false-mode result a “probability density function.” For a Poisson discrete distribution, the mathematically precise name is probability mass function (PMF)—the probability of an exact integer count. The documentation’s term should not be used to imply a continuous density, fraction of a count or an interval of nonzero width. For discrete counts, the cumulative result is the sum P(X=0)+…+P(X=x).

In the mathematical Poisson model:

P(X = k) = exp(−λ) × λ^k / k!, for integer k ≥ 0
P(X ≤ k) = Σ[j=0..k] exp(−λ) × λ^j / j!

The factorial and exponential identities are independent ways to validate sample arithmetic. Zoho separately documents EXP and FACT, but there is no need to reconstruct PMFs in a live worksheet if POISSON is correctly functioning. Special handling of λ=0 and huge count values remains a native testing question: Zoho only says λ<0 is invalid; it does not explicitly list all boundary results.

Assume a service team historically receives an average of three requests per hour, the rate is sufficiently stable, and requests are reasonably independent. We choose this as a hypothetical Poisson model, not an analysis of real requests. Set the following worksheet data:

Paste or enter values into A2:B7; place shown formulas in C2:D7. Predicted results, independently calculated and rounded to twelve digits:

Why the two columns differ: A two-request hour has exact probability around 0.224; an hour with two or fewer requests includes all possibilities with 0, 1 and 2 requests, totalling roughly 0.423. Adding every exact-count PMF from k=0 to 2 provides an independent check: 0.049787068368+0.149361205104+0.224041807655 ≈ 0.423190081127 (the displayed rounded terms can differ in the last decimal).

For the same mean of three requests per hour, suppose staffing is adequate for up to four requests, while five or more may require overflow support. The cumulative probability of up to four is:

=POISSON(4;3;TRUE)

Independent prediction: 0.815263244524. The overflow probability P(X≥5) is the complement:

=1-POISSON(4;3;TRUE)

Independent result: 0.184736755476, or about 18.47%. Note the off-by-one issue: “five or more” is 1−P(X≤4), not 1−P(X≤5). Similarly “more than five” means six or more, requiring 1−P(X≤5). Threshold phrasing must match the cumulative function’s inclusive count convention.

For the probability of three through six requests inclusive, use P(X≤6)−P(X≤2):

=POISSON(6;3;TRUE)-POISSON(2;3;TRUE)

Independent predicted result: 0.966491464691−0.423190081127 ≈ 0.543301383564. This formula relies on both terms using cumulative mode. Subtracting an exact probability from a cumulative one would not represent the stated interval.

Exposure scaling. If the mean is 3 requests per hour and the underlying rate is constant, the modelled expected number in a two-hour window becomes λ=6; for half an hour, λ=1.5. This is a mathematical rate/exposure conversion, not an instruction to blindly double observed counts. For a 30-minute window:

=POISSON(0;1.5;FALSE)

The independent expected result is exp(−1.5) ≈ 0.223130160148—the modelled probability of no arrivals in half an hour. Rate changes during the day, clustered arrivals or scheduled work invalidate the simple constant-rate model; in those situations modelling each period separately or using a different process may be more honest.

Parameterize your worksheet. Put the hourly average in H2=3, counts 0–10 in A2:A12, and the model in B2:

=POISSON(A2;$H$2;FALSE)

Fill down; if H2 changes, the expected probabilities change without formula rewrites. Add C2 =POISSON(A2;$H$2;TRUE) to compare exact versus cumulative behaviour. Avoid summing a truncated probability list and expecting exactly 1: if your table stops at count 10, the probability of counts 11 and above is omitted. Complete distributions sum to one over all nonnegative integers.

Alternative λ example. With five expected arrivals, =POISSON(2;5;FALSE) mathematically predicts 0.084224337489, while =POISSON(2;5;TRUE) predicts 0.124652019483. These are for a mean of 5, not 3; the shape of the mass function shifts as λ changes. A high observed count does not itself change the historical λ—estimating λ from data is a separate modelling decision.

Poisson versus binomial. The already documented BINOMDIST handles a fixed number of trials with a success probability. The Poisson model can approximate some rare-event binomial counts under appropriate assumptions but has no fixed maximum count. Compare BINOMDIST and POISSON only after expressing the same event/exposure question, and do not present a mathematical approximation as proof that Zoho’s two native functions return exactly equal decimals.

Poisson counts versus exponential waits. Under an ideal Poisson process, interarrival times are modelled exponentially. Zoho documents EXPONDIST for exponential probabilities, but the two functions answer different questions: counts per interval versus continuous time between events. You cannot use POISSON mode FALSE to calculate waiting-time density.

The official Zoho reference lists #NUM! for negative x or λ. It also lists generic #VALUE! for incompatible data types, #NAME! for a misspelled name, and #REF! for invalid cell references; its general #N/A! explanation uses lookup contexts and should not be relabelled a POISSON-specific failure. Its public text does not settle fractional-count coercion, omitted mode, λ=0 output, extreme numerical precision or multi-cell arrays. Z1

Further native tests: Exactly x=0/λ=0, decimal x=2.7 (do not guess whether it rounds), Boolean/empty/text inputs, missing third argument, localized punctuation, very high rates/counts, underflow in extremely unlikely tails and precise returned error strings. Zoho does not give a per-function minimum version, account-plan rule, or universal mobile/desktop compatibility matrix in this reference. The cloud-oriented English documentation is the verified scope of this manuscript.

Zoho documents POISSON(x;λ;mode) with semicolons; Microsoft’s Excel function is called POISSON.DIST(x,mean,cumulative), and Microsoft explicitly documents how Excel handles a non-integer x. That is other-product evidence only—it does not authorize saying Zoho truncates or rounds non-integer values. Spreadsheet migrations may rewrite names, separators and Boolean arguments; verify a simple exact and cumulative count in the target program. Do not assume that dotted POISSON.DIST is a Zoho-recognized alias without evidence. Z1 M1

• Z1 Zoho Sheet POISSON, exact syntax, mode semantics and documented errors, checked 2026-10-13: Zoho POISSON.
• Z2 Zoho Sheet BINOMDIST, related discrete binomial model, checked 2026-10-13: Zoho BINOMDIST.
• Z3 Zoho Sheet EXPONDIST, separate continuous waiting-time model, checked 2026-10-13: Zoho EXPONDIST.
• Z4 Zoho Sheet EXP, independent algebraic exponential relation, checked 2026-10-13: Zoho EXP.
• Z5 Zoho Sheet FACT, independent factorial relation, checked 2026-10-13: Zoho FACT.
• M1 Microsoft Support POISSON.DIST, comparison with Excel, not native Zoho behaviour; checked 2026-10-13: Excel POISSON.DIST.

Validation disclosure: Numerical predictions were computed independently in Python/SciPy and checked against the standard PMF formula and CDF summations. No Zoho Sheet account was accessed, no formula executed, no output/screenshots captured. Native review should record app/locale/date, all rows, zero and noninteger cases and actual error strings before making stronger claims.