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GAMMADIST in Apple Numbers: Gamma CDF versus density

Statistical Intermediate Apple Numbers

Model nonnegative measurements with gamma cumulative probability or density, with explicit mode.

Verification status: Official documentation referenced; example outcomes independently reasoned or arithmetically checked where stated. Not executed in Apple Numbers. How formula examples are checked

Quick answer: Apple’s =GAMMADIST(0.8,1,2,1) gives 0.329679953964361 (cumulative probability); changing the last argument to 0 gives 0.33516002301782 (density). The final argument is required by the published signature and selects two different questions. [A1, C1]

When is the gamma distribution appropriate?

A gamma distribution is a family of models for nonnegative values, often used for waiting times, aggregated positive quantities, or service-duration uncertainty. It is continuous and may be skewed. The function does not assess whether observed data genuinely follow that distribution or estimate alpha and beta from measurements. These examples assume a mathematical model and report what follows from it. [A1, C1]

Apple’s exact syntax, order, argument types and defaults

=GAMMADIST(x-value, alpha, beta, function-type)
Position Apple name Apple type Required Bound / interpretation / default
1 x-value Number value Yes Evaluation point, x ≥ 0; no default.
2 alpha Number value Yes Positive alpha parameter (alpha > 0); no default.
3 beta Number value Yes Positive beta parameter (beta > 0); no default.
4 function-type Modal value Yes 1/TRUE = CDF P(X≤x); 0/FALSE = PDF density height. No default documented.

A useful terminology caution: Apple’s own description calls both alpha and beta shape parameters. In the standard gamma parameterization independently checked against Apple’s two published numeric examples, alpha is the shape, while beta behaves as a scale (not a rate). For example with alpha=1 and beta=2, the CDF at x=0.8 is 1−EXP(-0.8/2) ≈ 0.329679953964361, matching Apple. Thus do not replace beta with its reciprocal from a differently parameterized reference. This scale inference is an independently reproduced mathematical interpretation, not a quotation of Apple’s parameter label. [A1, C1]

When alpha>0 and beta>0, the gamma CDF rises from zero toward 1 as x grows. The density is the derivative of this CDF, and can be greater than 1 for some concentrated models; it is not the probability of exactly x. The model’s mathematical mean is alpha×beta, and variance is alpha×beta². [C1]

Reproducible Numbers table: waiting-time uncertainty

Create a Numbers table called WaitingModel with at least eight columns. Set B1 Time x, C1 Alpha, D1 Beta, E1 CDF, F1 Density. Enter the input numbers below in B–D for rows 2–6, and copy the E2/F2 formulas down. All time units are hypothetical and should be tracked in headings, not appended into input cells. [A1, A5, C1]

Row B: x C: alpha D: beta E: formula (mode 1) Expected E F: formula (mode 0) Expected F
2 0 2 3 =GAMMADIST(B2,C2,D2,1) 0 =GAMMADIST(B2,C2,D2,0) 0
3 0.5 2 3 =GAMMADIST(B3,C3,D3,1) 0.0124379876276 =GAMMADIST(B3,C3,D3,0) 0.0470267624939
4 1 2 3 =GAMMADIST(B4,C4,D4,1) 0.0446249192349 =GAMMADIST(B4,C4,D4,0) 0.0796145900638
5 2 2 3 =GAMMADIST(B5,C5,D5,1) 0.144304801612 =GAMMADIST(B5,C5,D5,0) 0.114092693118
6 4 2 3 =GAMMADIST(B6,C6,D6,1) 0.384940011063 =GAMMADIST(B6,C6,D6,0) 0.117154283607

For alpha=2, beta=3, P(X≤4) is approximately 0.384940011063. The density at 4 is 0.117154283607 per time unit; it is not the probability of finishing at precisely x=4. With alpha=2, the density at 0 is mathematically zero; other positive alpha values below 1 have different behavior near zero, which needs domain-aware handling. [C1]

Apple’s own paired examples and independent checks

=GAMMADIST(0.8,1,2,1)
=GAMMADIST(0.8,1,2,0)

Apple publishes 0.329679953964361 for CDF and 0.33516002301782 for density. Independently, for alpha=1 and beta=2 the exponential special case gives 1−EXP(−0.8/2) → 0.329679953964361, while EXP(−0.8/2)/2 → 0.33516002301782. These match at Apple’s displayed precision; they were not evaluated in Numbers. [A1, C1]

Advanced applications

Interval probability between times 1 and 4

=GAMMADIST(4,2,3,1)-GAMMADIST(1,2,3,1)

Expected 0.340315091828. The function must be in cumulative mode 1 for both terms. Subtracting two densities, even if they are positive, is not an interval probability. [C1]

Right-tail time risk

=1-GAMMADIST(4,2,3,1)

Expected 0.615059988937, the modeled probability of a positive waiting time greater than 4. Extreme right-tail cancellation is possible under finite precision; Numbers-specific behavior for extreme arguments is untested. [C1]

Unit-scale comparison: scale beta doubles the model’s time unit

For a fixed alpha=2, changing beta from 3 to 6 doubles both the mean and every positive gamma quantile. But GAMMADIST(4,2,6,1) ≈ 0.144304801612, lower than CDF at x=4 for beta=3, because the same threshold is earlier in the broader distribution. Misreading beta as a rate would reverse this interpretation. [C1]

Convert a target percentile into a threshold

=GAMMAINV(0.9,2,3)
=GAMMADIST(GAMMAINV(0.9,2,3),2,3,1)

The quantile is 11.6691605096 and the second expression mathematically checks to 0.9, within numerical precision. Use GAMMAINV for an x cutoff and GAMMADIST for a probability or density; neither is a localized alias of the other. [A2, C1]

Locale-specific name and punctuation

In Apple’s Brazilian Portuguese help, the matching function is DISTGAMA. A locale equivalent of the official example is =DISTGAMA(0,8; 1; 2; 1), where the comma is a decimal separator and semicolons divide the arguments. This translates the same official help item; do not create a duplicate English route for DISTGAMA. Inspect the function name offered by the installation’s Formula Editor. [A3]

Numbers-specific references and formula entry

Build each example as a Numbers table, not an assumed Excel worksheet: enter row-1 headings, type the numeric values in the listed cells, enter each formula beginning with =, and fill down only where instructed. B2 addresses a cell in the current table; copied relative references change by row. A locked reference such as $C$2 keeps a shared parameter fixed. Apple documents that a sheet can contain multiple named tables; if the source is in another table named Model, select the actual cell in the Formula Editor or use a table-qualified reference such as Model::B2. Do not assume a single global sheet grid. [A5]

For LOGINV and GAMMAINV, keep probability inputs as numeric values strictly between 0 and 1, not 90 for a 90% target. You may format cumulative probability results as Percent, but formatting does not alter a mathematical value. The LOGNORMDIST and GAMMADIST first arguments are observation thresholds, not probabilities. Keep parameters numeric and do not append units inside formula input cells. If copied formulas point to the wrong table or a moving shared parameter, inspect the reference tokens in Numbers. [A5]

Errors, input limits and troubleshooting

Apple’s individual function help specifies permitted argument domains; it does not document a stable displayed error token for every invalid input, device, locale, or cell-type conversion. Consequently the following are predicted invalid inputs or user misunderstandings, not observed Numbers errors. No #... token or screenshot is invented. [A1, A4]

Input or symptom What’s wrong Remedy
=GAMMADIST(-1,2,3,1) Apple requires x ≥ 0. Correct the negative observation/model or choose another model.
=GAMMADIST(1,0,3,1) Alpha must be strictly positive. Supply alpha >0.
=GAMMADIST(1,2,-3,1) Beta must be strictly positive. Supply beta >0; verify scale-versus-rate convention.
=GAMMADIST(1,2,3) Apple’s listed fourth argument is required, with no default documented. Add 1 (CDF) or 0 (density).
Density value wrongly formatted as 60% probability at x PDF is a curve height, not single-point probability. For percent chance at or below x, use mode 1.
Calculations disagree with a rate-form gamma textbook Apple’s examples align with beta as scale. In a rate-form model λ, use beta=1/λ only if mathematically appropriate.
All parameters are constant but copied formulas vary References to parameter cells moved. Use locked $C$2 and $D$2 for shared parameters, or fill per-row inputs deliberately.
Spreadsheet language changes function acceptance Localized spelling and separators differ. Use the locale Functions Browser and check numeric cells.

Compatibility, support and evidence boundaries

The function’s English signature and examples are from Apple Formulas and Functions Help 15.4, inspected 2026-10-13 UTC. Apple also documents formula editing and references for Numbers; the Help navigation covers Apple’s iWork formula ecosystem, but the existence of a current documentation article is not proof of native execution on Mac, iPad, iPhone, iCloud, or older versions. Apple publishes app release documentation, which must not be read as evidence that this particular function debuted in that release. Its earliest supported Numbers version was not independently established. [A1, A4, A5, A6]

Do not transfer Excel, Google Sheets or other spreadsheet engines’ default arguments, error codes, statistical parameter definitions, locale punctuation, or precision guarantees to Numbers. Finite-precision corner cases, text/blank coercion, large-parameter stability, and exact error messages require native confirmation. These guides were checked against documentation and by independent numerical computation only; there was no Apple Numbers execution, import/export test, screenshot or historical-platform regression test. Intermediate expected results below are rounded for readability; a Numbers cell’s number format may show fewer digits. [A1, C1]

  • GAMMADIST — complete official signature and both published CDF/PDF outcomes. [A1]
  • GAMMAINV — inverse gamma cumulative probability. [A2]
  • EXP — exponential component used in the alpha=1 mathematical check. [C1]
  • LN — natural logarithm is useful when analysing positive-data models; this is not an Apple-documented default argument. [C1]

Claim-level official references and validation sources