PHI Function (Calligra Sheets)

• Introduced: First release not established; the name is present in Calligra 2.4.0 source.
• Supported in: Calligra Sheets 26.08.2, with no retirement established.
• Scope: Desktop Sheets with the Statistical module. Examples use English semicolon-separated formulas.
• Checked: October 14, 2026. Selected formulas ran through the official 25.04.2 engine library. Source and independent mathematics were checked. This is engine evidence, with no GUI-cell or import verification; 26.08.2 was source-checked only.

PHI returns the standard normal density φ(x)=exp(−x²/2)/√(2π). It is a density height, not the lower-tail normal probability. Its maximum occurs at zero, where the value is approximately 0.398942280401. Function help.

=PHI(x)

Supply one real numerical argument. There is no cumulative flag. Positive and negative inputs are both meaningful, and symmetry gives φ(−x)=φ(x). Registration and implementation.

The engine and mathematical columns are separate. Results are rounded to twelve decimal places, or thirteen significant digits in scientific notation for very small or large values. This is a display choice, not an accuracy guarantee. NaN and infinity are nonfinite outputs; an engine error flag is checked separately from those values.

For a normal model with mean μ=10 and standard deviation σ=2, the density at y=12 is φ((12−10)/2)/2=φ(1)/2≈0.120985362260. Dividing by σ is essential when moving from a standardized density to the original measurement units. This is an independently computed model example; the listed PHI formulas are the executed engine cases.

The probability at one exact point of a continuous distribution is zero, even though the density there is positive. To calculate an interval probability, use a cumulative normal calculation at the two endpoints and subtract.

The lognormal-density alternative on the companion LOGNORMDIST guide uses PHI after standardizing a logarithm. Its division by xσ is a change-of-variable factor, not an arbitrary scaling.

At x=1, φ(x)≈0.241970724519. The executed symmetry check subtracts φ(−1) and returns zero. A local small-interval approximation, probability≈φ(x)×width, is only an approximation; for width 0.1 around x=1 it gives about 0.024197072452, not an exact interval integral.

The executed advanced check was:

=PHI(1)-PHI(-1)

Its engine result was 0.

The source uses a stored normalization constant and floating-point exponential arithmetic. Very large |x| can underflow even though the mathematical density is positive. The selected ±10 cases remain nonzero. Nonnumeric text “abc” was converted to zero and returned the peak density; do not treat that plausible result as proof of valid input. Missing or extra arguments produce #VALUE!.

• Function-specific implementation
• Versioned function help
• Phi helper
• Conversion helper

The statistical module and value-calculation helpers are byte-identical between the inspected 25.04.2 and 26.08.2 sources. That comparison supports the source discussion but does not establish later-version native output, every locale or arbitrary extreme-parameter accuracy.

NORMSDIST, NORMDIST.