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BETAINV in Apple Numbers: percentile thresholds

Statistical Intermediate Apple Numbers

Use the beta inverse CDF to find thresholds, with strict probability conditions, scaling and examples.

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 =BETAINV(0.99,1,2,0,1) returns 0.9: in that mathematical beta model 99% of the distribution is at or below 0.9. The function goes from a probability to a threshold x; Apple’s separate BETADIST runs in the opposite direction. Both shape parameters must be positive, and the requested probability must satisfy 0 < p < 1, not include 0 or 1. [A1]

What question BETAINV answers

Use BETAINV when you have a hypothetical beta model for a bounded measurement and need a quantile: “What cutoff is exceeded by only 10% of this model?” BETAINV(0.9,alpha,beta,lower,upper) targets the 90th percentile. The output shares the units of x, not units of probability. It does not infer alpha or beta from observations and is not the same thing as an automatically calculated confidence interval. [A1, C1]

Official syntax, ordering, scalar argument types and defaults

=BETAINV(probability, alpha, beta, x-lower, x-upper)
Position Apple’s exact name Apple type Required? Documented condition Default
1 probability Number value Yes 0 < p < 1 None
2 alpha Number value Yes alpha > 0 None
3 beta Number value Yes beta > 0 None
4 x-lower Number value Optional ≤ x-upper 0
5 x-upper Number value Optional ≥ x-lower 1

Apple expresses the inequality for bounds using ≤, but a nondegenerate continuous beta distribution requires upper > lower in the usual mathematical definition. This review does not assert how Numbers responds to coincident bounds. Both optional arguments can be left off to use [0,1]; when you need a different upper bound, supply both explicit endpoints instead of assuming a locale-specific empty-argument placeholder. [A1, C1]

Common input mistake: 90% represented as a numeric percentage is 0.9; a bare number 90 is not a valid probability. Conversely, output 0.9 might be a cutoff x=0.9, not a 90% probability when the bounds are not 0 and 1. [A1, C1]

Create a complete Apple Numbers percentile table

Create a Numbers table BetaQuantiles with headings B1 Target cumulative p, C1 Alpha, D1 Beta, E1 Lower, F1 Upper, G1 Cutoff x. Enter the row values below into B–F. Put =BETAINV(B2,C2,D2,E2,F2) in G2 and fill through G8; format G as Number. All expected results are independently calculated. [A1, A2, C1]

Row B: p C: α D: β E: lower F: upper Formula in G Expected x
2 0.1 2 2 0 2 =BETAINV(B2,C2,D2,E2,F2) 0.391600211318183
3 0.25 2 2 0 2 =BETAINV(B3,C3,D3,E3,F3) 0.652703644666139
4 0.5 2 2 0 2 =BETAINV(B4,C4,D4,E4,F4) 1
5 0.75 2 2 0 2 =BETAINV(B5,C5,D5,E5,F5) 1.34729635533386
6 0.9 2 2 0 2 =BETAINV(B6,C6,D6,E6,F6) 1.60839978868182
7 0.25 1 2 10 20 =BETAINV(B7,C7,D7,E7,F7) 11.3397459621556
8 0.75 1 2 10 20 =BETAINV(B8,C8,D8,E8,F8) 15

Interpretation: with α=β=2 on [0,2], the 90th percentile is approximately 1.6084 (not a probability); roughly 10% of the modeled continuous scores exceed it. Rows 7–8 illustrate that a model on [10,20] produces thresholds in that interval even though the input p remains strictly between 0 and 1. The row-2 result is an Apple-published example. [A1, C1]

Validate Apple’s actual examples independently

=BETAINV(0.5,1,2,0.3,2)     → 0.797918471982869
=BETAINV(0.99,1,2,0,1)      → 0.9
=BETAINV(0.1,2,2,0,2)       → 0.391600211318183

Each arrow is Apple’s documented output; independent inverse-beta calculations reproduce all three to stated precision. This does not mean these formulas were typed into Numbers or tested on a device. [A1, C1]

Practical and advanced analyses

1. Invert a cumulative probability: round-trip check

=BETADIST(BETAINV(0.75,2,2,0,2),2,2,0,2)

Expected mathematical result 0.75, within floating-point precision. This asks for an x threshold with CDF 0.75, then runs the CDF to verify that target. The two function identifiers are different and have separate Apple help pages. [A4, C1]

2. See which quantiles are symmetric

When alpha=beta=2, lower=0, upper=2, the mathematical 25th and 75th percentiles sum to 2. Using the table values, 0.652703644666139 + 1.34729635533386 ≈ 2. This is a distributional symmetry check; it does not hold unchanged for unequal alpha and beta. [C1]

3. Use a closed-form beta family as an independent check

For α=1, β=2 on [L,U], the mathematical CDF is 1−(1−t)^2 where t=(x−L)/(U−L). Thus the inverse is x=L+(U−L)*(1−√(1−p)). At p=0.75, L=10, U=20, this gives 10+10*(1−0.5)=15, agreeing with row 8. These identities explain the result; they are not asserted as Apple’s numerical algorithm. [C1]

4. Find a modeled upper-alert boundary

Suppose a simulated score is supported on 0–2 with α=β=2. You want the upper 5% alert boundary, so choose a CDF target of 0.95, not 0.05:

=BETAINV(0.95,2,2,0,2)

Expected ≈ 1.72929927566. Calling this a “95% confidence limit” would overstate what the model does; it is a modeled 95th percentile, conditional on assumed parameters. [C1]

5. Understand how changing the bounds scales outputs

Compare =BETAINV(0.5,2,2,0,1) (expected 0.5) with =BETAINV(0.5,2,2,0,2) (expected 1). Only the interval width doubled; the distribution shape and percentile did not change. Do not try to fix a units mismatch by altering alpha or beta arbitrarily. [C1]

Errors, domain validation and troubleshooting

Input / interpretation Issue Correction
=BETAINV(0,2,2) Probability is strictly above zero. Choose a p in (0,1).
=BETAINV(1,2,2) Probability is strictly below one. Do not assume the edge limit is accepted.
=BETAINV(95,2,2) Entered 95 instead of 0.95. Use 0.95 or a true 95%-formatted numeric cell.
=BETAINV(0.5,0,2) Alpha must be >0. Supply positive alpha.
=BETAINV(0.5,2,-1) Beta must be >0. Supply positive beta.
=BETAINV(0.5,2,2,2,1) Lower > upper violates Apple’s bounds. Order endpoints correctly.
=BETAINV(0.5,2,2,1,1) Degenerate mathematical interval, response not established. Choose a genuinely positive interval width; native test before claiming behavior.
=BETAINV(B2:B8,2,2) Apple documents scalar probability input. Calculate one row at a time.
Result confused with a probability Result is a cutoff x, not the CDF. Evaluate the returned x with BETADIST.

Compatibility and alias safety

Apple’s English 15.4 catalogue lists the identifiers BETAINV and BETADIST separately. This review does not assert a dotted BETA.INV spelling as an Apple alias, nor prove how Excel implements a similar formula. The .15.4 help path identifies a documentation snapshot; absent a function-specific Apple release note, it is not evidence of the exact version this function first appeared. probability, alpha, beta, optional bounds and the strict p-domain are evidenced directly by Apple. [A1, A4]

Entering these formulas in an Apple Numbers table

Create the named table specified in the examples, type the headings in row 1, and enter numbers as numeric cells, not as pasted strings containing explanatory units. In a formula inside that same table, B2 refers to its B-column row 2; use TableName::B2 when referring from a different table and choose the actual reference token through the Formula Editor if you rename a table. A Numbers sheet can contain several independent tables: this is not an Excel-like single sheet-wide cell grid. Apple’s reference guidance explicitly documents table-qualified references. [A2]

English official examples put commas between arguments and use a dot for decimal numbers. The installed locale may use different function translations, decimal marks, or argument separators; no localized alias of BETAINV is verified in this review. Choose BETAINV from the formula browser and reproduce the displayed syntax in your own locale. Do not create another English guide/URL simply because a translated name is discovered. Apple describes the distinction between a number value (scalar argument) and an array or range; the following function-specific scalar arguments have not been shown by Apple to automatically consume entire ranges. [A1, A3]

How to interpret the numbers: Expected outputs below are separately computed using mathematical identities and Python/SciPy/mpmath , or transcribed and separately checked from Apple examples. They are not screenshots, native Apple Numbers execution results, or tested platform behaviour. Display precision can vary with formatting, rounding, locale and software implementation. [C1]

Compatibility, editions and validation limits

This guide documents the English Formulas and Functions Help 15.4 reference, checked 2026-10-13 (UTC). It does not establish behaviour in older Numbers releases, other spreadsheet programs, all devices, or every regional edition. The math checks establish expected values under the documented inputs, but native Numbers execution and round-trip evaluations remain NOT TESTED. The 15.4 help document is evidence of the signature in that documentation edition, not by itself a historical minimum-version claim. [A1]

No Excel or Google Sheets name or error treatment is imported as Apple Numbers evidence. In particular, old-version compatibility, exact overflow thresholds, implicit text-to-number coercion, handling of blank cells, display precision and localized parse rules require native testing before publication. All cited official Apple links were checked 2026-10-13 and are listed below. [A1–A3]

Practical troubleshooting (not observed error codes)

The function’s official domain rules are authoritative. The invalid formulas described here are predicted violations of those documented rules, not executions in Numbers; the docs do not give a guaranteed displayed #... token for each failure. First check whether cells are numeric, then arguments and bounds, then table references and the application’s locale. Do not substitute IFERROR to silently hide a model-validation failure. If a computed result seems wrong, reproduce one literal Apple example exactly before converting it into parameterized table cells. [A1–A3]

Sources and validation

Native-validation limitation: No Numbers execution or screenshots were performed. Relevant locale, precision and input-coercion edge cases require testing in the target app.