PHI Function in Gnumeric: Formula, Examples, and Troubleshooting

=PHI(x)

There are no optional arguments or defaults. The 1.12.62 release introduces PHI; a prior Gnumeric release should not be assumed to recognize it. Upstream Gnumeric 1.12.62 statistical source declares a one-number signature and computes the standard normal density through dnorm(x,0,1,FALSE); its helper text calls this the standard normal distribution density. Upstream test registration is basic and is not evidence that these tutorial examples ran.

The defining formula is:

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

The displayed formula is a mathematical substitute that also works in software with compatible EXP/SQRT/PI functions. The equal sign and decimal dot below reflect the English formula examples; some locales change separator conventions. PHI itself has a single argument and needs no comma.

1. Type z-score in A1 and normal density in B1.
2. Fill A2:A6 with the five z-scores below.
3. Enter =PHI(A2) into B2 and copy down through B6.
4. Format results to 10 decimal places.

Explanation: The standard normal curve peaks at zero and is symmetrical. Therefore PHI(1)=PHI(-1). The result 0.39894 at zero is a density measured per unit of z-score, not a 39.9% chance of observing exactly zero (for a continuous distribution, the probability of one exact point is zero).

Suppose a process measurement is approximately normal with mean 100 units and standard deviation 15 units. Put a measurement 115 in A2, the mean 100 in B2 and the standard deviation 15 in B3. Enter:

=PHI((A2-$B$2)/$B$3)/$B$3

Expected result: approximately 0.0161313816 density per measurement unit, because (115-100)/15 = 1, PHI(1) ≈ 0.2419707245, and dividing by 15 converts from z-score units to measurement units. This is the normal density, not the probability of falling between two measurement values. To calculate a probability over a range, integrate via a cumulative distribution function (such as NORMDIST with its appropriate options) or subtract two CDF values.

The $ absolute references keep the fixed mean/standard-deviation cells unchanged as the formula is copied for more observations. Confirm B3 is positive before applying the formula.

• PHI versus GAUSS: PHI gives curve height; GAUSS gives signed area between the mean and a z-score. They are not substitutes.
• PHI versus NORMSDIST: NORMSDIST gives the left-tail cumulative probability, ranging from 0 to 1; PHI is a density, peaking at 1/SQRT(2*PI()).
• Wrong scale: PHI expects a standardized input. For a mean of 100 and SD 15, using 115 directly instead of z-score 1 produces a misleading value near zero.
• Output displayed as zero: For large absolute z-scores the density is extremely small and may appear zero due to number formatting or underflow. Change number format and check the input.
• Input errors: Ensure that the input is a number and the software version supports PHI. Text cells or incomplete formulas can produce an error; precise Gnumeric error codes were not confirmed by native execution. Do not copy Excel-specific error messages as guaranteed Gnumeric behaviour.
• Invalid process SD: The advanced formula divides by B3, so zero SD is mathematically undefined; it is not a PHI argument error, but a problem in the surrounding formula.

Microsoft Excel documents PHI(x) for its recent versions. Where PHI is unsupported, calculate =EXP(-A2^2/2)/SQRT(2*PI()) with the target application’s equivalent functions. For nonstandard normal density, scale the standard density by 1/standard_deviation; do not omit this factor. Other spreadsheet programs may differ in function aliases, accepted arguments and numeric precision. This page does not assert native LibreOffice, Numbers or Excel tests were run.

• NORMDIST — cumulative/density computation with a mean and spread.
• NORMSDIST — standard normal cumulative value.
• STANDARDIZE — standardize raw measurements.
• GAUSS

The release-matched statistical-function source diff contains the help_phi documentation and C call dnorm(x,0,1,FALSE); the registered function has one numeric argument and basic upstream test status. See the official release announcement naming PHI. These are documentation/source inspections rather than application execution.

• Gnumeric — function documentation
• Gnumeric official function reference

Verification status: Official documentation checked; example outcomes were independently reasoned or arithmetically checked where applicable. Not executed in Gnumeric.

• No genuine Gnumeric screenshots are included; this guide does not use mock application images.