NORMDIST in Apple Numbers: Normal CDF vs Density

Choose NORMDIST when you have an assumed normal distribution and need either a cumulative percentage for a threshold or the corresponding curve density. Common modelling uses include production measurements, forecast-error models and standardized-score demonstrations, when the normality assumption is reasonable. It does not automatically determine whether observed data are normal, estimate population parameters, or count actual observations; values here are probabilities from a model. For percentile cutoffs use NORMINV instead. [A1, C1]

=NORMDIST(num, average, stdev, function-type)

The fourth argument changes the kind of result: 1 is P(X ≤ num) and 0 is f(num), a density in inverse units of the measurement. A density is not a probability at a single real-valued point. The distinction is fundamental because a bell-curve peak is the height of the curve, and depending on standard deviation that height can be larger than 1. Apple’s documented mode choices are exactly those shown above; this guide does not assert unsupported coercion for arbitrary text or 2. [A1, A2, C1]

For a normal model with mean μ and positive standard deviation σ, write z = (x − μ)/σ. The cumulative probability is Φ(z) and the density is φ(z)/σ with φ(z) = EXP(-z²/2)/SQRT(2×PI()). This is a mathematical identity, not an Apple implementation specification or a claim that a non-Numbers application behaves the same. The inverse NORMINV(p,μ,σ) yields x at cumulative probability p. [C1]

Create a table named NormalModel with at least six columns (A–F), entering Value x in B1, Mean in C1, Population SD in D1, CDF in E1 and Density in F1. Put the following numeric values in rows 2–6, then enter the two formulas shown in row 2 and fill down; no arrays, hidden templates or external sheets are required. Do not type units into the numeric cells. [A1, A3]

For row 4, the observation equals the model mean: cumulative probability is 0.5, and density is about 0.03989422804 per measurement unit. Row 5 at 60 corresponds to one positive standard deviation, giving cumulative probability about 0.841344746069. The 0.84 means approximately 84% under the assumed model, not 84% of the actual measurements already entered. [C1]

If all observations share one mean and SD, put mean 50 in C2 and SD 10 in D2, then in a separate output cell use =NORMDIST(B3,$C$2,$D$2,1). Copying that formula holds parameters fixed; without dollar signs the references would move. Apple’s reference help explicitly distinguishes relative and absolute references. [A3]

Apple gives =NORMDIST(22,15,2.5,1) → 0.997444869669572 and =NORMDIST(22,15,2.5,0) → 0.00316618063319199. Standardizing first gives z=(22−15)/2.5=2.8. Independent calculation confirms Φ(2.8)=0.997444869669572 and φ(2.8)/2.5=0.00316618063319199 at the quoted precision. This mathematical check and Apple’s reference are separate from any native test. [A1, C1]

Probability of falling between two measurements

With a model mean=50, stdev=10, the probability between 40 and 60 is a difference of cumulative probabilities:

=NORMDIST(60,50,10,1)-NORMDIST(40,50,10,1)

Expected: 0.682689492137 (68.27%). Use two CDFs, not two density heights. The endpoints have no mass individually under a continuous normal model, so inclusive versus exclusive makes no difference to this theoretical interval. [C1]

Probability above a threshold

=1-NORMDIST(65,50,10,1)

Expected: 0.0668072012689 (about 6.68% above 65 under this model). This is the complement of the CDF. If the threshold is very many standard deviations above the mean, subtracting a rounded CDF from 1 can lose floating-point precision. Whether a particular Numbers version exhibits a particular extreme-tail rounding is untested. [C1]

Linking observed z-scores with standard-normal references

=NORMDIST(60,50,10,1)
=NORMSDIST((60-50)/10)

Both have the independently calculated expectation 0.841344746069. Apple explicitly documents the equivalence of NORMDIST with a mean of 0, SD of 1 and cumulative mode to NORMSDIST. The formulas standardize into a z-score using simple arithmetic, and the second is a mathematical cross-check rather than an instruction to duplicate a new page for an alias. [A1, A10, C1]

When the fourth argument is 0

=NORMDIST(50,50,10,0) has density 0.0398942280401. Change the model SD to 0.1 and the curve peak becomes about 3.98942280401; this is a valid density, not an impossible probability. Do not use 1−NORMDIST(x,μ,σ,0) to find an upper tail—that subtracts a density from 1. [C1]

Apple lists input domains for these functions but does not enumerate the precise displayed error token for every invalid input, version, imported document, or type conversion. Therefore the examples below describe the cause and corrective action, not an invented #... error string. [A1, A4]

For P(X≤x) use NORMDIST(…,1); for curve height at x use NORMDIST(…,0); for x corresponding to target percentile p use NORMINV(p,μ,σ); and when mean=0 and SD=1, the corresponding cumulative standard-normal function is NORMSDIST(z). These are different questions even when some numeric results happen to coincide in special cases. [A1, A10, C1]

The worksheets above assume formulas and their source data live in the same Numbers table, so B2 identifies the first input cell under a row-1 header, and $B$2 stays fixed if copied. Unlike a one-sheet-only model, Numbers documents multiple named tables per sheet. If you move the calculation to another table, the cross-table pattern is NormalModel::B2 (when the source table is named NormalModel). A sheet-qualified form is Sheet 2::NormalModel::B2. These conventions come from Apple’s cell-reference guide, not an Excel workbook model. Use the Formula Editor to select the actual cell and avoid accidental name collisions. [A3]

The worksheet in this tutorial uses scalar formulas per row, not a dynamic-array spill convention. Enter row-1 headers first, enter numeric values in source columns, put the formula into the indicated output cell, then fill it down. If copy/fill moves a parameter that should be fixed, use a fixed reference like $C$2, or explicitly refer to the same source row. Numbers documents the difference between A1 and $A$1; copying a formula may change the former but not the latter. [A3]

This page uses the English function name NORMDIST and English-language examples with a period as decimal point and a comma between arguments. Apple’s Brazilian Portuguese 15.4 help for the same function instead titles the corresponding function DIST.NORM and demonstrates syntax with the locale-appropriate separator; an independently checked example is =DIST.NORM(22; 15; 2,5; 1). This is a verified localized name tied to the same official help identifier. [A9]

Copying an English formula verbatim to a Portuguese-Brazil installation can therefore fail or be interpreted differently. Inspect the insertion suggestion in that installation’s Functions Browser and match the spreadsheet’s language/region settings. In particular, a comma in 0,89 is a decimal separator in the Brazilian example while semicolons separate arguments. The B2:C6 colon in a range is neither of those separators. Apple documents the general Functions Browser on Mac/iPhone/iPad. [A5, A9]

The reference version used here is Apple Formulas and Functions Help 15.4 (English, checked 2026-10-13 UTC). Apple lists the 15.4 Numbers release on Mac and iPhone, and the current release notes specify macOS 26 or later for Numbers 15.4 on Mac and iOS 26 or later on iPhone. Those are app-level requirements, not proof that this particular function was tested on either device, available in every earlier version, or available identically in Pages, Keynote and iCloud. [A5, A6, A7]

These functions are documented in Apple Formulas and Functions Help 15.4; the release notes do not establish the earliest version that introduced this specific function. Do not convert the current documentation-version number into an unsupported first-supported-version claim. [A1, A6, A7]

A different spreadsheet may use a similarly named normal-distribution function or different punctuation and defaults. This article makes no claims about Excel, Google Sheets, or other spreadsheet engines, and it does not silently treat their behavior as evidence for Numbers. Where the function page omits a property (for example exact error strings, limits for extreme finite values, coercion of blank/text cells, or internal floating-point precision), this guide deliberately marks it unverified rather than inventing it. [A1, A4]

Validation boundary: Every displayed numeric expectation was independently calculated in a Python/SciPy statistical calculation environment using the displayed formulas and inputs; this is not native execution in Apple Numbers. The official examples were also compared with the published values. No screenshot, Numbers file, import/export conversion, formula-error display, or cross-platform execution test was produced. Roundoff and the number of decimals shown in a Numbers result cell depend on numeric calculation/display choices; compare sufficiently many digits before assuming a mistake. [C1]

• NORMDIST: exact syntax and published examples — this page’s authoritative English reference. [A1]
• NORMDIST — general normal cumulative probability or density selected explicitly by mode. [A10]
• NORMINV — inverse of the general normal CDF. [A10]
• NORMSDIST — standard normal cumulative probability (not a density). [A10]
• NORMSINV — inverse of the standard normal CDF. [A10]
• PHI — standard normal density at one z-score. [A10]

• [A1] Apple — NORMDIST (Formulas and Functions Help 15.4) — function name, purpose, signature, type/domain, named examples, help-linked relationships; checked 2026-10-13.
• [A2] Apple — Types of arguments and values — Apple definitions of numeric/modal arguments (where the individual function specifies a modal choice); checked 2026-10-13.
• [A3] Apple — Refer to cells in formulas — B2, B2:B6, table and sheet :: notation, relative and absolute references; checked 2026-10-13.
• [A4] Apple — Review formula errors and best practices — general formula debugging, not a source of invented exact errors; checked 2026-10-13.
• [A5] Apple — Formulas and Functions Help welcome — Functions Browser and iWork app context; checked 2026-10-13.
• [A6] Apple — What’s new in Numbers on Mac — Numbers 15.4 release and OS requirement; checked 2026-10-13.
• [A7] Apple — What’s new in Numbers on iPhone — Numbers 15.4 release and OS requirement; checked 2026-10-13.
• [A8] Apple — New functions in Numbers on iPad — names PHI specifically among current new statistical functions; checked 2026-10-13.
• [A9] Apple — Portuguese-Brazil help for DIST.NORM — function translation and locale-specific formula example; checked 2026-10-13.
• [A10] Apple — linked normal-distribution family pages — independently checked each of the five named official Apple help URLs on 2026-10-13; these establish relative purpose without asserting spreadsheet cross-engine compatibility.
• [C1] Mathematical calculations using independent Python/SciPy (scipy.stats.norm) and standard normal PDF/CDF identities; using the displayed example values. Never a native Numbers run.