BETADIST in Apple Numbers: bounded cumulative probabilities

• Introduced: Documented in Apple’s Formulas and Functions Help 15.4; the original introduction release is not established.
• Availability: Documented for Numbers in Apple’s Formulas and Functions Help 15.4; no final supported-version endpoint is established.
• 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 BETADIST reference. Documentation and independent arithmetic checks; no native Numbers test.

The beta family models a continuous value on a finite interval, often a hypothetical proportion between 0 and 1 or a scaled quantity between two bounds. The positive shape parameters alpha and beta govern the model’s profile. Unlike GAMMADIST, which has a separate PDF/CDF mode, Apple’s BETADIST description specifically says cumulative beta distribution probability value and its published syntax contains no mode argument. A cumulative probability answers “what fraction is at or below x?”

This mathematical distribution is not a substitute for validating whether real data obey a beta model. BETADIST does not estimate alpha/beta from your sample, nor does it report a confidence interval for a binomial success proportion without an explicitly chosen statistical method. The guide provides numeric examples under assumed parameter values.

=BETADIST(x-value, alpha, beta, x-lower, x-upper)

The published argument sequence matters: the upper/lower arguments are positions 4 and 5, not alternative shape parameters. The usual mathematical continuous beta distribution requires a nonzero interval width (upper > lower), though Apple’s paragraph separately states only the less-than-or-equal comparisons against x. Do not infer that a zero-width range has a meaningful beta CDF from those inequalities. Avoid undocumented empty placeholders when wanting a custom upper bound with an omitted lower; supply both bounds in that case.

Create a Numbers table named BetaCases with at least eleven columns and eight rows. Use B1 Threshold x, C1 Alpha, D1 Beta, E1 Lower, F1 Upper, G1 Cumulative probability. Enter numeric rows 2–8 below, and type =BETADIST(B2,C2,D2,E2,F2) into G2. Fill it down. The scale in these rows is a synthetic bounded score on 0–2, not a claim about observed business performance.

Interpretation: with α=β=2 and lower=0, upper=2, 50% of the modeled scores are ≤1 and another 50% >1. The shape is symmetric about 1. Format G-column cells as Number or Percent as desired; 0.5 is 50% after formatting, not “0.5%.” One original Apple example at x=0.1 exactly matches the table.

=BETADIST(0.5,1,2,0.3,2)    → 0.221453287197232
=BETADIST(1,1,2,0,1)        → 1
=BETADIST(0.1,2,2,0,2)      → 0.00725

The displayed outputs above come from Apple’s official help; our independent mathematical evaluation reproduces them within shown precision. Apple’s displayed number of spaces after commas is cosmetic in these examples, but decimal/argument punctuation still depends on the selected locale.

1. The interval probability is a difference of two CDFs

=BETADIST(1.5,2,2,0,2)-BETADIST(0.5,2,2,0,2)

Expected probability 0.6875—68.75% of the modeled scores are between 0.5 and 1.5. Do not add both CDFs or call either result a probability density. This assumes a continuous distribution, for which individual exact points have zero probability.

2. With symmetric alpha=beta=2, verify the analytic CDF

For an interval normalized to t=(x-lower)/(upper-lower), the mathematical CDF is 3t²−2t³. If α=β=2, lower=0, upper=2, x=1.5, then t=0.75, so 3×0.75²−2×0.75³ = 0.84375. This independently checks a value without relying solely on a software library. This identity is mathematical, not Apple’s published implementation.

3. Check the two optional-bound defaults explicitly

=BETADIST(0.25,2,2)
=BETADIST(0.25,2,2,0,1)

Both are expected 0.15625 because Apple gives lower=0 and upper=1 as defaults. Do not copy this result to a domain with endpoints 0 and 2 unless the bounds are explicitly supplied.

4. A simple one-sided threshold alert

Keep the practice table unchanged. Put the numeric fraction 0.6 in J2 and enter the following formula in K2:

=BETADIST(J2,2,2)

=BETADIST(0.6,2,2) gives 0.648. Do not interpret this as “64.8% confidence that the model is correct”—it is a CDF conditional on the model and parameters.

5. Invert a percentile using a different Apple function

=BETADIST(BETAINV(0.9,2,2,0,2),2,2,0,2)

Mathematically expected near 0.9. BETAINV computes the x cutoff given a probability, while BETADIST computes a probability given x. The outer result is susceptible to floating-point rounding. See the BETAINV guide for the inverse calculation.

The current English reference uses the undotted name BETADIST. This is not a deprecation or retirement notice. Its presence in 15.4 does not prove the same availability, names or outputs for every earlier Mac/iPhone/iPad/iCloud release. Changing bounds is a linear rescaling of the same beta shape: a standard-beta 0.75 becomes x=1.5 on [0,2]. Neither percent-cell formatting nor changing the display to 2 decimal places changes this underlying model.

Enter the model inputs as numeric cells. A same-table reference such as B2 differs from a cross-table reference such as BetaCases::B2; Apple’s cell-reference guide explains qualified table names. Use the actual table name when adapting the examples.

The examples use decimal points and comma-separated arguments. Apple’s function syntax guidance specifies semicolon-separated arguments when the decimal separator is a comma. Choose the function through Numbers’ formula editor when adapting the syntax to your locale.

Expected values were independently recalculated using analytic beta identities, exact arithmetic and numerical inverse checks. Coincident bounds, blank/text coercion, overflow limits and localized aliases are not established by these calculations. The invalid-input table identifies documented domain conflicts; it does not promise an exact displayed error token for every case.