PHI Function (OpenOffice Calc)

• Introduced: First release not established; documented by OpenOffice.org 3.3.
• Supported in: Apache OpenOffice 4.1.16; this is a verified release, not a final supported version.
• Scope: Desktop Calc, English function names and semicolon-separated formulas. Locale and file-format mappings may differ.
• Checked: October 14, 2026. Eight reference formulas and the listed boundary cases were recalculated in 4.1.16 on Linux x86-64, Build 9816, using generated native ODF-formula input. Source and independent mathematics were also checked.

PHI returns the standard-normal probability density φ(z)=EXP(−z²/2)/SQRT(2×PI()). It is a curve height, not a lower-tail probability and not the probability of one exact observed value. Release implementation.

=PHI(Z)

Exactly one numeric standardized value is used. Positive and negative values are accepted. The density is even, so PHI(−z)=PHI(z). Its maximum occurs at zero and is approximately 0.398942280401. The direct exponential implementation can underflow to zero for sufficiently large magnitudes.

These eight formulas were recalculated in the stated 4.1.16 build using generated native ODF-formula input. Saved numeric results agree with independent high-precision calculations; values below are rounded to twelve decimal places. The comparison concerns stored values, not a claim that every display format prints the same digits.

At z=1, =PHI(1) gives approximately 0.241970724519. At z=−1 it gives the same height. Put standardized cutoffs in J2:J3 and use =PHI(J2) with clear labels such as “standard-normal density”; formatting these values as cumulative percentages would be misleading.

For a normal measurement with mean 70 and standard deviation 8, its density at x=82 is =PHI((82-70)/8)/8. The independently calculated value is approximately 0.016189699458 per measurement unit. The final division by 8 is required when rescaling a density.

For a continuous normal model, the probability of an interval is the area under the curve. A narrow interval near z=1 with width 0.02 has the midpoint approximation 0.02×PHI(1), approximately 0.004839414490. This is an approximation; the exact interval probability is obtained from a cumulative difference such as NORMSDIST(1.01)−NORMSDIST(0.99).

The density ratio PHI(2)/PHI(1) equals EXP(−1.5), approximately 0.223130160148. The common normalization factor cancels. This ratio compares heights, not cumulative tail probabilities.

A log-density calculation can preserve information when the density itself is too small: the mathematical standard-normal log density is −z²/2−LN(2×PI())/2. Exponentiating it immediately can recreate the original underflow.

A stored zero at a very large input does not prove an impossible event. The exact normal density remains positive for every finite real argument, but ordinary floating-point arithmetic cannot represent arbitrarily small positive numbers.

PHI and GAUSS do not have interchangeable outputs. Nor should PHI be confused with a cumulative normal function simply because some mathematical texts use a visually similar capital phi symbol. Use the worksheet function’s documented definition.

The following boundary formulas were recalculated in the same named method. H5 and H6 are deliberately empty cells where referenced. An error display remains an error even if the underlying ODS numeric attribute stores zero.

Tested import limitation

The separate XLSX input produced an unresolved _xlfn.PHI identifier and #NAME? for these cases. Those results identify an import-path failure, not the native function’s calculation. The successful results above came from generated ODF formulas followed by hard recalculation; they were not manually typed.

• NORMSDIST
• NORMDIST
• STANDARDIZE

• Function-specific release entry
• Numeric helper or argument collection

The retained input and saved ODS preserve the tested formula identities. A stale SUM cache changed from 999999 to 3, and an uncached SUM returned 7 after hard recalculation. Native evidence is limited to the eight reference formulas and listed boundary cases in this exact release and method. Composite explanatory examples are independent calculations unless listed among those cases; other releases, locales and extreme-parameter accuracy were not established.