WEIBULL in Apple Numbers: Reliability, CDF and Density

Apple documents the two-parameter Weibull distribution. It is useful for a model of positive lifetimes, component failure ages, or other continuous nonnegative measurements, provided a fitted shape and scale are justified by data. Let x be elapsed time, alpha the shape and beta the scale. The standard mathematical cumulative failure probability and probability density are:

CDF(x) = 1 - EXP(-POWER(x / beta, alpha))
PDF(x) = (alpha / beta) × POWER(x / beta, alpha-1) × EXP(-POWER(x / beta, alpha))

Here alpha affects how the shape of the failure-time distribution changes with age and beta sets the time scale. With alpha=1 the mathematical model reduces to an exponential distribution; alpha>1 has increasing hazard, and 0<alpha<1 decreasing hazard. Those hazard interpretations are mathematical properties of the model, not independent Apple execution observations. [A1, C1]

=WEIBULL(x-value, alpha, beta, function-type)

Both parameters must be positive; Apple does not document an implicit default for any of the four arguments. Beta is a scale, not an inverse rate; changing beta to 1/beta changes the model. The CDF is between zero and one, while a density has units of inverse time and can exceed one for some valid parameters. [A1, C1]

Create a Numbers table named FailureModel with B1 Time x, C1 Cumulative failed by x, D1 Density at x, F1 Shape alpha, and G1 Scale beta. Set F2 to 2 and G2 to 4. Enter time values 0, 1, 2, 3, 4, 6, 8 in B2:B8. In row 2 enter these formulas and fill down:

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

The references $F$2 and $G$2 are fixed within this Numbers table, as Apple’s cell-reference help describes. In another table, the equivalent parameter references can be qualified with FailureModel::F2 and FailureModel::G2. [A3]

For this particular alpha=2 table, the density at zero is mathematically zero. Do not generalize that statement to every shape: with 0<alpha<1, density diverges as x approaches zero and the exact behavior of WEIBULL(0,alpha,beta,FALSE) requires native testing. All table values are independent mathematical expectations, not captured Numbers cells. [C1]

1. Reliability: probability a part lasts beyond time 3

If a fitted model uses alpha=2, beta=4, the survival probability is 1-CDF(3):

=1-WEIBULL(3,2,4,TRUE)

Expected 0.569782824731. WEIBULL(3,2,4,TRUE) alone is the complementary failure-by-3 probability 0.430217175269. The survival calculation does not prove the part is safe or that failure-time data truly fit a Weibull model. [A1, C1]

2. Hazard rate at a time point

The continuous-model hazard is the density divided by survival probability, not the raw density itself:

=WEIBULL(3,2,4,FALSE)/(1-WEIBULL(3,2,4,TRUE))

Expected 0.375 per time unit because for a two-parameter Weibull, hazard(x)=(alpha/beta)×(x/beta)^(alpha−1). Do not multiply density by 100 and label it the probability of an exact-time failure. [C1]

3. Calculate a 90th-percentile lifetime from the parameters

The distribution’s inverse CDF is x=beta×POWER(-LN(1-p),1/alpha) for 0<p<1. Numbers documents POWER and LN separately. For shape 2, scale 4, target cumulative probability .9:

=4*POWER(-LN(1-0.9),1/2)

Expected 6.069708517541 time units. Checking the mathematical CDF at this x returns 0.9. This is a calculated quantile formula, not a claim that Apple offers a separate WEIBULLINV function. [A6, C1]

4. Link with the exponential distribution when shape is 1

For alpha=1 and beta=2, WEIBULL(3,1,2,TRUE) is mathematically 0.776869839852, the same CDF as an exponential waiting-time model with rate 0.5 per unit. Apple’s separate EXPONDIST help uses a rate-style parameter. This identity is a math comparison, not an assertion of identical rounding or every-edge-case behavior across both Numbers functions. [A6, C1]

5. Interpret a density correctly

WEIBULL(1.5,0.5,2,FALSE) is about 0.1214225 per time unit. A probability for a short interval around 1.5 depends on integrating density over its width; a continuous random variable has probability zero at any one exact time under the standard model. The TRUE result is the probability of a lifetime no greater than 1.5. [A1, C1]

Apple’s article does not enumerate a universal error string, display rounding rules, overflow/underflow thresholds or the behavior of the mathematical singularity. Leave these unverified until tested in the target Numbers version; do not import assumptions from another program. [A1, C1]

Does TRUE return reliability? No. TRUE returns the cumulative probability of failure by x for the lifetime model. Reliability is the complement 1-WEIBULL(x,alpha,beta,TRUE).

Can a WEIBULL density be greater than 1? Yes for some mathematical parameter combinations because it has units of inverse x. Probability for an interval comes from area under that density, not its height.

Are alpha and beta interchangeable? No. Apple describes alpha as shape and beta as scale; swapping them changes distributional meaning.

• EXPONDIST for an exponential waiting-time distribution, mathematically related when shape is one. [A6]
• LN for natural logarithms used in the inverse-quantile calculation. [A6]
• POWER for exponentiation in the inverse-quantile calculation. [A6]

These are direct official Apple 15.4 function links, not claims that the corresponding tutorial paths have been approved or deployed. [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, =WEIBULL(1,5;0,5;2;TRUE) rather than =WEIBULL(1.5,0.5,2,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, WEIBULL, Formulas and Functions Help 15.4, https://support.apple.com/guide/functions/weibull-ffa6f16a1e8/15.4/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.