PHI Function in Google Sheets: Evaluate the standard-normal density
Learn PHI in Google Sheets with explicit inputs, worked scenarios, independently checked expectations and troubleshooting.
Verification status: See the evidence and test scope described in this guide. How formula examples are checked
Quick answer: Evaluate the standard-normal density with PHI. The Google PHI documentation supplies the contract summarized here.
Syntax and meaning
=PHI(x) takes a point on the standard-normal axis. Its result is the density at that point for mean 0 and standard deviation 1. Google explicitly identifies it with NORMDIST using mean 0, standard deviation 1 and cumulative FALSE.
A density is a curve height. It is not the probability of a value being below x, and it is not the probability of observing exactly x. The mathematical reference used for the checks below is exp(-x²/2) / sqrt(2π).
Worked example: center and tails
Enter -2, 0 and 2 in A2:A4. In B2 enter =PHI(A2) and copy through B4. Independently calculated expected results are approximately:
| x | Density |
|---|---|
| -2 | 0.0539909665 |
| 0 | 0.3989422804 |
| 2 | 0.0539909665 |
The equal tail heights follow from squaring x in the density formula. The center has the greatest height. Do not read 0.3989422804 as a 39.9% cumulative probability at zero.
Advanced scenario: standardize a measurement
A synthetic measurement model has mean 100 and standard deviation 15. An observation of 130 is two standard deviations above the mean. =PHI((130-100)/15) therefore has expected result approximately 0.0539909665, the density on the standardized axis.
For a density in the original measurement units, the change of scale matters: =PHI((130-100)/15)/15 is approximately 0.0035993978 per measurement unit. This second expression follows mathematically from transforming a standard-normal variable; PHI itself still accepts only the standardized point. A point-density calculation does not establish that real observations follow a normal distribution.
Troubleshooting
- For an accumulated probability or interval probability, choose the appropriate distribution calculation instead of treating a density as an area.
- If the input is in original units, first establish the intended mean and positive standard deviation. PHI does not accept those as extra arguments.
- Keep enough decimal places to inspect small densities; a displayed zero may just be rounding.
- Extreme-tail underflow, text conversion and native error tokens have not been verified here.
Related guide
See normal-distribution guide when the mean, standard deviation or cumulative mode needs to be supplied directly. The equivalence above is documented by Google; no other application’s behavior was executed.
Availability and version history
Documented in the rolling Google Sheets service. Documentation reviewed October 19, 2026; an introduction date or numbered release was not established. Examples use English function names, decimal points and comma separators; adapt separators to the sheet’s locale.
Verification scope and sources
The official reference establishes the syntax and function-specific contract. Worked values are original, independently derived expectations, not native Google Sheets observations. No account access, UI screenshot or cross-version execution was performed. Native behavior outside the documented scope remains unverified.