PHI in Collabora Office: Evaluate Standard-Normal Density
Use Collabora PHI for the standard-normal density, understand symmetry, and distinguish curve heights from cumulative probabilities.
Verification status: See the evidence and test scope described in this guide. How formula examples are checked
Quick answer: =PHI(0) has independently expected result approximately 0.398942280. It is the height of the standard-normal density at zero, not a 39.9% probability of observing exactly zero.
Syntax and density definition
=PHI(Number)
Number is the standardized evaluation point. The dedicated Collabora help identifies PHI as a probability density and equates it to NORM.S.DIST(Number;FALSE()). It also gives symmetric examples at 2.25 and −2.25.
The independent mathematical expression is EXP(-(Number^2)/2)/SQRT(2*PI()). The standard normal distribution has mean zero and standard deviation one. Its density is positive for finite real inputs, symmetric about zero and largest at zero.
A continuous probability comes from an area under the density curve across an interval. A curve height at one point is not that interval’s probability. Do not substitute PHI where a formula asks for a normal cumulative probability.
Example 1: build a symmetric density table
Put −2, −1, 0, 1, 2 in A2:A6. Enter =PHI(A2) in B2 and fill down. Independent targets are approximately 0.053990967, 0.241970725, 0.398942280, 0.241970725 and 0.053990967.
In C2 enter =EXP(-(A2^2)/2)/SQRT(2*PI()) and fill down. Compare B and C as a mathematical definition check. The squared input explains why opposite signs produce the same density.
The official inputs 2.25 and −2.25 both have independent result approximately 0.031739652, which the help rounds to 0.0317. That equality does not mean their left-tail cumulative probabilities are equal.
Example 2: contrast density with cumulative probability
At z=1, PHI(1) is approximately 0.241970725. The standard-normal left-tail probability at the same point is approximately 0.841344746, obtained with =NORM.S.DIST(1;1) or mathematically 0.5+GAUSS(1).
At z=−1, the density is unchanged but the left-tail probability is approximately 0.158655254. A density curve is symmetric; the cumulative curve increases from zero to one.
Label columns “Density” and “Cumulative probability.” Applying a percent format to every numeric output can conceal the fact that the two columns have different meanings.
Advanced application: obtain an original-scale normal density
For a normal model with mean mu and positive standard deviation sigma, its density at x is PHI((x-mu)/sigma)/sigma. The division by sigma is essential because it adjusts the horizontal measurement scale.
With x=130, mu=100 and sigma=15, the standardized score is two. =PHI((130-100)/15)/15 has independently expected result approximately 0.003599398 per original unit.
Using PHI(2) alone would give a density per standardized unit instead. Probabilities remain invariant under a consistent unit transformation, but densities change with the scale of the measurement axis.
Advanced application: approximate a narrow interval
Near the center, an interval of width 0.1 centered on zero has rough midpoint approximation =0.1*PHI(0), independently approximately 0.039894228.
The exact model probability for the interval from −0.05 to 0.05 is =2*GAUSS(0.05), independently approximately 0.039877612. The small difference is approximation error: the density varies across the interval rather than staying perfectly flat.
This example explains how a density becomes an area, but it is not a recommendation to use coarse rectangles for precise probability calculations. Wider intervals or steeply changing parts of the curve need more care.
Errors and troubleshooting
PHI accepts a standardized numeric evaluation point in these examples. The brief help does not specify all native input coercions or extreme-value underflow behavior. A tiny tail density displayed as zero may be a numerical or formatting issue, not a statement that the mathematical density is exactly zero.
If an original-scale density looks too large, check the missing division by sigma. If a result is expected to represent a left-tail probability, use the cumulative form instead. The function does not fit a normal distribution or validate the assumptions of a model supplied elsewhere.
Availability and verification
The exact function is documented in Collabora Office 26.04 desktop help. The snapshot was retrieved on 2026-10-13 and hash-verified on October 15, 2026. Desktop help is not proof of Online or mobile availability. First introduced release and last functioning version are unknown; the dedicated source does not establish them.
No native Collabora or WPS execution occurred. Numerical values are independent mathematical expectations, not certification of application errors or all numerical limits. Formulas use English names and semicolon separators.