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PHI in Apple Numbers: Standard Normal Density

Statistical Intermediate Apple Numbers

Graph and interpret the standard-normal probability density with Apple Numbers PHI: reproducible z-table, symmetry and limits.

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 =PHI(0) returns 0.398942280401433, the height of the standard-normal density at z=0. At z=1 or -1, PHI returns 0.241970724519143 in Apple’s examples. PHI is a density function. It is not P(Z≤z), which requires the separate Apple function NORMSDIST. [A1, A8, A10, C1]

Which statistical question PHI answers

PHI evaluates the standard-normal probability density φ(z) at a given standardized score z. The curve has mean 0 and standard deviation 1, peaks at z=0, and is symmetric: φ(-z)=φ(z). A density is a curve height per unit z, not the probability at an exact point. For a continuous model, an exact value has probability zero; the probability of a range is an area under the density. [A1, C1]

A common spreadsheet error is to label a PHI result as “probability below the threshold.” It is not that. Use NORMSDIST for the cumulative probability and subtract two CDF values for the area between thresholds. Apple’s own 15.4 Functions Help provides both functions as separate identifiers and its “What’s new” page explicitly includes PHI as a new statistical function. [A8, A10]

Exact function name and argument order

=PHI(z-value)
Position Argument Type Required Accepted values / default
1 z-value Number value Yes A numeric z-score; no other finite-domain restriction or optional default is documented.

The PHI function accepts a single numeric argument, not a mean or SD, and Apple does not document a cumulative/density modal argument. A z-score is standardized with the normal mean 0 and SD 1. Its mathematical expression is φ(z)=EXP(-z²/2)/SQRT(2×PI()). This identity is independent mathematical reasoning; it does not assert the internal algorithm used by Numbers. [A1, A2, C1]

Reproducible Numbers worksheet: plot a bell-curve density

Create a table named DensityCurve with z-score in B1 and Density φ(z) in C1. Enter the five numeric z-values below into B2:B6, put =PHI(B2) in C2, and copy the formula down. Plot the B and C columns as an XY scatter/line chart if you choose to visualize the mathematical shape; that chart will display density height, not the probabilities in a frequency sample. A native Numbers chart was not created or validated as part of this review. [A1, A3, C1]

Row B: z Formula in C Independent expected density C
2 -2 =PHI(B2) 0.0539909665132
3 -1 =PHI(B3) 0.241970724519
4 0 =PHI(B4) 0.398942280401
5 1 =PHI(B5) 0.241970724519
6 2 =PHI(B6) 0.0539909665132

The table is symmetric around z=0: rows for z=-1 and z=+1 have equal density, as do z=-2 and z=+2. The highest recorded height is at zero, approximately 0.398942280401. The density diminishes moving away from the mean even though the cumulative probability grows from zero toward one across increasing z. [C1]

Apple’s published examples and direct mathematical confirmation

Apple’s 15.4 function page explicitly gives three examples: =PHI(0) → 0.398942280401433, =PHI(1) → 0.241970724519143, and =PHI(-1) → 0.241970724519143. Independent evaluation of the standard-normal density agrees to Apple’s displayed precision. The formula’s symmetry is both Apple-documented and mathematically derived here. [A1, C1]

Advanced uses: distinguish density, probability and scaling

PHI(1) compared with the cumulative function

=PHI(1)
=NORMSDIST(1)

The independent expected results are respectively 0.241970724519 (density height) and 0.841344746069 (area left of 1). These numbers are not interchangeable. Do not treat 0.24197 as a “24.2% chance of z being exactly 1,” or 0.84134 as a density. [A10, C1]

Compute a probability between two z cutoffs correctly

=NORMSDIST(1)-NORMSDIST(-1)

Expected 0.682689492137, approximately 68.27% of the standard normal distribution between -1 and +1. A shortcut such as PHI(-1)+PHI(1) has no probability interpretation for the interval; summing two heights is not finding the area beneath the curve. For a numerical approximation by discrete integration you would need a width and a sufficiently fine grid, but the CDF difference is the direct expression for this model. [C1]

Evaluate a density for a nonstandard normal model

Suppose a measurement model has mean 50, population SD 10 and observed value x=60. The standardized z is (60−50)/10=1, and the general-model density requires a scaling by 1/SD:

=PHI((60-50)/10)/10
=NORMDIST(60,50,10,0)

Both independently yield 0.0241970724519 per measurement unit. It would be incorrect to use =PHI((60-50)/10) alone as that result, because it has density units per standardized z, not per 10-unit measurement scale. [A10, C1]

Estimate relative density heights for two standardized scores

=PHI(2)/PHI(0)

Expected 0.135335283237. The height at z=2 is roughly 13.5% of the peak height at z=0. This is a ratio of density values, not a probability that z=2 occurs or a probability that z lies beyond 2. Apple’s PHI page does not promote the ratio as a statistical test; it is simply mathematical interpretation of the density formula. [C1]

Version-aware model sharing

PHI is named in Apple’s “New Functions” list for the latest Numbers release; if a collaborator reports that a copied PHI formula is not recognized, first confirm the exact installed app release and language before investigating numbers. Apple’s Apple Numbers 15.4 guides and Function Help establish current support context, but only an actual workbook execution can confirm behavior on that collaborator’s device. [A1, A8]

Errors and misconceptions

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]

Input or symptom Meaning What to do
=PHI() One required z-value argument is missing. Enter a numeric z-score.
=PHI(1,TRUE) Apple documents one argument, without a mode. Remove the extra argument; use NORMSDIST for cumulative probability.
=PHI(1,0,1) This function is not NORMDIST with mean/SD. Use NORMDIST for a specified nonstandard distribution.
Density result is described as “24.2% likelihood of exactly 1” A density height is not point probability. Use differences of NORMSDIST for interval probabilities.
Plot appears reversed or values repeat A copied relative reference may point to the wrong cell. Check B2, B3, … and positive/negative symmetry.
Unrecognized formula in an older shared Numbers file PHI is explicitly named as new in current release notes, but old-version behavior was not tested. Check installed version and device compatibility.
Unexpected text coercion, precision or a specific error token Apple does not enumerate every case here. Confirm numeric cells and consult general formula error guidance.

A correct PHI density can be greater than 1 in the scaled, nonstandard normal model after dividing by sufficiently small σ; the standardized PHI peak itself is about 0.39894. A value over 1 is not evidence of a probability error when the result is a density. [C1]

Numbers-specific cell references and reproducibility safeguards

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 DensityCurve::B2 (when the source table is named DensityCurve). A sheet-qualified form is Sheet 2::DensityCurve::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]

Locale-specific spelling and separators

This page uses the English function name PHI 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 PHI and demonstrates syntax with the locale-appropriate separator; an independently checked example is =PHI(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]

Compatibility, version, and evidence limits

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]

Apple explicitly lists PHI among the new 15.4 statistical functions on its iPad “What’s new” page. The 15.4 function page agrees with that release context; do not claim it was present in earlier versions without separate evidence. [A1, A8]

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]

  • PHI: 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]

Official references and claim-level verification (2026-10-13 UTC)