POISSON in Apple Numbers: Event Counts, PMF and CDF

• Introduced: Documented in Apple’s Formulas and Functions Help 15.4; the original introduction release is not established.
• Availability: Currently documented for Numbers in that help version; it is not a final supported-version endpoint.
• Scope: Shared iWork function help. Availability in individual clients and older releases is not established by this check.
• Checked: October 14, 2026, against the official POISSON reference. Documentation check; not a new native test.

The Poisson model counts occurrences in a fixed interval when arrivals can reasonably be treated as independent and the average rate per interval is stable. Examples include incoming requests per hour, faults per square metre, and calls per 15-minute period. The average, usually written λ, must use the same time/area unit as the event count. For nonnegative integer k, the mathematical model has point probability

P(X=k) = EXP(-lambda) × lambda^k / k!

and cumulative probability P(X≤k)=Σ(j=0..k)P(X=j). These equations are independent statistical definitions; Apple’s help documents how its POISSON selector chooses between the two outcomes. The statistical model is not justified merely because the data happen to be counts: clustered events, trends, caps, or dependence need different analysis. [A1, C1]

=POISSON(events, average, function-type)

Use a whole, nonnegative event count and a meaningful nonnegative rate for this mathematical distribution. Importantly, this particular Apple help entry does not explicitly publish numeric validation bounds, truncation rules for fractional events, or the treatment of zero-average edge cases. The recommendations here concern sound inputs, not a promise about how Numbers accepts or rejects unusual values. All three arguments are shown in Apple’s documented signature; there is no stated default for function-type. [A1]

Create a Numbers table Arrivals with header labels: B1 Event count k, C1 Exactly k, D1 At most k, F1 Expected events λ. Enter 3 in F2 and whole k values 0..7 in B2:B9. In C2/D2 enter the formulas below and fill downward:

C2: =POISSON(B2,$F$2,FALSE)
D2: =POISSON(B2,$F$2,TRUE)

The fixed reference $F$2 prevents the parameter from shifting when formulas are filled. In a formula written in a different Numbers table, qualify the parameter as Arrivals::F2 when required; Apple documents the TableName::Cell convention. [A3]

The CDF should never decrease as k increases; the difference between adjacent CDF values should equal the PMF of the later k. The table’s last CDF is below 1 because there is still positive probability of eight or more events. All numbers above are independently calculated and have not been produced by a native Numbers session. [C1]

1. Four or more arrivals during a period

=1-POISSON(3,3,TRUE)

Expected 0.352768111218. The complement of P(X≤3) is P(X≥4). Do not subtract the CDF at 4 if you mean four or more; that would exclude the exactly-four case. [A1, C1]

2. Between two and five arrivals inclusive

=POISSON(5,3,TRUE)-POISSON(1,3,TRUE)

Expected 0.716933784497. CDF(5) minus CDF(1) leaves counts 2,3,4,5. The right subtraction boundary is one below the desired lower bound. [A1, C1]

3. Correctly convert the observation interval

Suppose an observed process averages 6 events/hour and you want the probability of zero in 15 minutes. Fifteen minutes is one-quarter hour, so the corresponding mean is 1.5 events/15 min, not 6:

=POISSON(0,1.5,FALSE)

Expected 0.223130160148, provided the rate is sufficiently stable and the assumptions are defensible. Entering POISSON(0,6,FALSE) would model a full hour’s zero-count probability rather than a 15-minute window. [C1]

4. Cross-check Apple’s published eight-event case

=POISSON(8,10,FALSE)

Expected 0.11259903214902, independently obtained from e^-10 × 10^8 / 8! and SciPy’s Poisson PMF. This agrees with Apple’s example but is not an Apple Numbers execution test. [A1, C1]

Apple’s article does not supply an exhaustive precision limit, fixed rounding rule, or specific error string for each invalid input. Do not transplant #NUM! behavior or other implementation details from Excel. [A1, C1]

Can POISSON return a probability density? Its FALSE mode returns discrete probability mass for exactly k events, not continuous density. The CDF mode adds masses for k or fewer.

Is mean 3 the same as exactly three arrivals? No. A process with mean 3 can produce 0, 1, 2, 3 or many more arrivals; the distribution assigns probabilities to each count.

Does Numbers call this POISSON.DIST? The current English Apple documentation explicitly calls it POISSON, with no period in the canonical name. Do not generate a second slug for another application’s spelling. [A1]

• EXPONDIST treats continuous waiting times, not discrete counts; it is mathematically related under a suitable Poisson-process model but cannot replace POISSON’s count PMF. [A6]
• BINOMDIST models a fixed number of independent binary trials, not an unconstrained count during an interval. [A6]

Apple’s English Formulas and Functions Help, edition 15.4, documents the function and its parameter order. Apple’s current Mac Numbers version history describes release 15.4, but does not identify this particular function among the features newly introduced in that release. The 15.4 documentation therefore cannot establish the earliest version that supports it. Apple currently states that Numbers 15.4 for Mac requires macOS 26 or later and Numbers 15.4 for iPad requires iPadOS 26 or later; availability, edge-case evaluation and precision on an actual Mac, iPhone, iPad or iCloud session have not been tested here. Do not treat Excel, Google Sheets or Python behavior as proof of Numbers compatibility. [A1, A4, A5]

The formulas here use the English decimal point and comma-separated arguments. Apple states that in a setting using the decimal comma you must separate function arguments with semicolons: for example, =POISSON(3;2,5;TRUE) rather than =POISSON(3,2.5,TRUE). This is a separator rule; it does not verify localized translations of function names or every regional input format. Apple says function names are case-insensitive. [A2]

Numbers uses named tables within sheets, with table-qualified cell references. A formula can refer to B2 in its own table, or to an explicitly qualified cell such as Model::B2 in another table, according to Apple’s cell-reference guidance. $ anchors hold fixed cells when filling formulas. [A3]

• [A1] Apple, POISSON, Formulas and Functions Help 15.4, https://support.apple.com/guide/functions/poisson-ffa59e0419/15.4/web/1.0. Checked 2026-10-13 UTC. Establishes function name, argument order, documented types/modes/bounds and Apple’s example; it does not certify historical app versions, native test outcomes or undocumented coercion.
• [A2] Apple, Functions overview, documentation edition 15.4. Checked 2026-10-13 UTC; establishes case-insensitive names, parenthesized arguments, required-versus-optional tokens and decimal-comma semicolon separator rule.
• [A3] Apple, Refer to cells in formulas, documentation edition 15.4. Checked 2026-10-13 UTC; establishes cell/range notation, cross-table :: qualifiers and Numbers-specific table references.
• [A4] Apple, What’s new in Numbers on Mac, checked 2026-10-13 UTC; version history describes Numbers 15.4 and macOS requirements. It does not individually date the introduction of this function.
• [A5] Apple, What’s new in Numbers on iPad and Formulas and Functions Help, checked 2026-10-13 UTC; establishes the current broader documentation/version context without establishing an individual iPhone/iCloud regression result.
• [A6] Apple, related references listed below are directly linked to their corresponding official 15.4 function help and checked 2026-10-13 UTC. They explain comparisons, not proof of another application’s behavior.
• [C1] Independent mathematical verification, 2026-10-13 UTC: local Python standard-library math evaluated against separately implemented open-source SciPy distribution calculations. Expected results are not native Numbers outputs or screenshots.

Native-validation limitations: No Numbers execution or screenshots were performed. Historical device compatibility, localized behavior and undocumented coercion/precision require testing on the intended platform.