PHI Function (LibreOffice Calc)

=PHI(Number)

PHI has one required argument, with no optional arguments or documented defaults. In the English Calc UI the function name is PHI. For a literal numeric value, use the decimal separator appropriate to your locale; the examples here use English-style decimal points. LibreOffice allows localized function names and separately configurable formula argument separators. PHI itself has only one argument, but formulas below using NORM.S.DIST, ROUND, etc. use Calc’s common semicolon (;) separator. If your Calc is configured for commas, adapt the separators; never replace decimal marks mechanically. LibreOffice Formula Options (accessed 2026-10-12).

The defining mathematical formula is:

phi(z) = EXP(-z^2/2) / SQRT(2*PI())

This density is always nonnegative and is symmetric: PHI(-z) = PHI(z). The largest density occurs at zero. These properties follow from the formula and were sampled independently and in native Calc, but should not be confused with a cumulative probability.

These examples use LibreOffice Calc 25.2.3.2 (Linux). Displayed numbers are rounded for readability; Calc stores further precision. The figures below are recorded in the saved evidence dated 12 October 2026.

Working with a cell reference: Put 1 in A2, then enter =PHI(A2) in B2. Expected output: approximately 0.241970724519143. Change A2 to -1 and the result is unchanged. In the saved test workbook, the density table begins in A2 and covers values from -4 to 4. The native test’s own referenced-input case uses its existing numeric A9 cell, which contains 1.

Why PHI is not a probability-of-exact-value function

Continuous distributions assign probability to intervals (areas under the curve), not to an exact point. For example, PHI(1) is about 0.24197, but this does not say that there is a 24.197% probability of a random observation being exactly 1. Use a cumulative distribution function to calculate the area over a range. The LibreOffice documentation identifies NORM.S.DIST(z;FALSE()) and NORMDIST(z;0;1;FALSE()) as the density equivalents of PHI. Official PHI Help; NORM.S.DIST Help.

For the probability that a standard normal value falls between 0.9 and 1.1, enter:

=NORM.S.DIST(1.1;TRUE())-NORM.S.DIST(0.9;TRUE())

The expected result is approximately 0.0483940644 (independently calculated using the standard normal CDF; this specific interval formula was not separately executed in the native test suite). The area is not PHI(1), which is a density at only one point. The difference of the cumulative probabilities calculates the entire interval’s probability.

Suppose a reference process is modeled as a normal distribution with mean 5 units, standard deviation 2 units, and you want the density at an observed measurement of 7 units. Standardize the observation first, then scale the standard-normal density by the standard deviation:

=PHI((7-5)/2)/2

Expected result: approximately 0.120985362259572 per unit. Native Calc executed the algebraically identical formula =1/2*PHI((7-5)/2) and returned 0.12098536225957168. Both expressions are equal by arithmetic; the exact displayed precision may differ by formatting. To make this reusable, enter the observation in B2, the mean in E1 and the positive standard deviation in E2:

=PHI((B2-$E$1)/$E$2)/$E$2

If B2=7, E1=5 and E2=2, the expected result is the same. $E$1 and $E$2 keep the distribution parameters fixed if the formula is copied down. Ensure E2 is positive; a zero standard deviation causes division by zero. This density calculation assumes a normal model and makes no claim that real data follow that model.

Compare the density and the cumulative area at the same z-score

The following two formulas deliberately answer different questions:

=PHI(1)
=NORM.S.DIST(1;TRUE())

The first gives density 0.2419707245. The second gives cumulative probability 0.8413447461 (area to the left of 1). Native Calc verified that the two values differ, and that =PHI(1)-NORM.S.DIST(1;FALSE()) returns zero. For a normal density with other mean or standard deviation values, consider NORM.DIST instead of manually repeating the transformation.

These are observations from LibreOffice Calc 25.2.3.2, not guaranteed error-code behavior for every Calc release or locale:

Troubleshooting checklist: Check that Number is a real numeric value (rather than unclean imported text), the decimal mark agrees with your locale, there is exactly one argument, and any dependent calculations are error-free. For a probability between two cutoffs, switch to cumulative NORM.S.DIST differences rather than PHI. A PHI result above zero is a density per unit of z, not an observation count. If you standardize raw measurements first, check that the standard deviation is greater than zero and that measurement units are consistent.

A subtle mistake with summing density values

Adding the heights PHI(-2)+PHI(-1)+PHI(0)+PHI(1)+PHI(2) gives approximately 0.990865662466096, as native Calc confirmed. This is not exactly a probability: it is a sum of curve heights at arbitrarily chosen points, with no explicit interval widths or integration method. To estimate the area under a density from sampled points, you need a numerical integration rule and appropriate step sizes. For a normal model in a spreadsheet, using the documented cumulative function is usually simpler and more accurate.

The underlying mathematics agrees across these products; do not assume identical numeric-text coercion, error codes, locale formatting, or floating-point rounding without testing the target application. See Microsoft’s PHI reference and Google’s PHI reference (checked 2026-10-12). No claim is made here about a separate OpenFormula namespace or the exact oldest LibreOffice version of PHI, because that information was not established from the PHI help entry checked for this guide.

• NORM.S.DIST — choose FALSE() for standard-normal density or TRUE() for the cumulative probability.
• NORM.DIST — model a normal distribution with its own mean and standard deviation.
• STANDARDIZE — convert a raw measurement to a z-score before calling PHI.

The following source pages were checked again on 2026-10-13. Their precise sections support the statements in this tutorial:

1. LibreOffice 26.8 Help: Statistical Functions Part Four — PHI. Defines PHI(Number), explains standard-normal density, gives PHI(0), PHI(±2.25) examples, and states density equivalence with NORM.S.DIST and NORMDIST.
2. LibreOffice 26.8 Help: Statistical Functions Part Five — NORM.S.DIST. Defines the Cumulative flag’s density versus cumulative behavior.
3. LibreOffice 26.8 Help: Formula Options. Supports the statements about localized names and configurable argument separators.
4. Microsoft Support: PHI function. Documents Excel’s availability, argument and density semantics. Documentation-only comparison.
5. Google Docs Editors Help: PHI function. Documents Google’s PHI syntax and meaning. Documentation-only comparison.

Verification status: Official documentation checked; 17/17 corrected predefined native assertions passed using LibreOffice Calc 25.2.3.2 on Linux. Nine additional input/error/coercion cases were observed, not pre-asserted. Independent reference densities were computed using exp(-z²/2)/sqrt(2π). The original first test run contained an incorrect test expectation that an unweighted sum of five density heights should exceed 1; the observed value was 0.9908656624660955. The test expectation was corrected to the independently calculated five-term sum, and the native test was rerun with all 17 checks passing. The saved initial mismatch and corrected script/results agree with this correction. No genuine application screenshot is included; no mock application images were created. Microsoft Excel and Google Sheets were not executed.