PHI in Excel: Standard-Normal Density and Curve Heights

=PHI(x)

The required argument is a standardized position on a distribution with mean 0 and standard deviation 1. The independent mathematical model used for the examples is exp(−x²/2)/sqrt(2×pi).

PHI has one argument, so it cannot directly accept a different mean or standard deviation. A raw measurement must first be standardized if it belongs to a normal model with different parameters.

Enter -2, -1, 0, 1 and 2 in A2:A6. Fill down:

=PHI(A2)

Expected values are 0.053990967, 0.241970725, 0.398942280, 0.241970725 and 0.053990967. Equal distances from zero have equal density. The largest of these values is at the mean.

These are curve heights. A height of 0.398942280 at zero does not mean a 39.8942280% chance of drawing exactly zero. Under a continuous normal distribution, every single exact point has probability zero even where the density is positive.

=PHI(0)
=NORM.S.DIST(0,TRUE)
=NORM.S.DIST(0,FALSE)

Expected outputs are 0.398942280, 0.5 and 0.398942280. The first and third describe density; the middle expression describes the cumulative area through zero.

This is a useful check when a worksheet was built from a table of standard-normal areas. A density column cannot be used as a percentile column simply by applying percentage formatting.

Suppose a fictional normal model has mean 100 and standard deviation 15. At measurement x=115, calculate:

=PHI((115-100)/15)/15

Expected density: 0.016131382 per measurement unit. Standardizing gives z=1, and dividing the curve height by the standard deviation restores the original horizontal scale.

Using PHI((115-100)/15) without the final division gives approximately 0.241970725, which is a density per standardized unit. It is not the same density on the original measurement scale. Always require a positive standard deviation.

For a narrow interval of total width 0.02 centered at z=1, density times width provides a simple approximation:

=PHI(1)*0.02
=NORM.S.DIST(1.01,TRUE)-NORM.S.DIST(0.99,TRUE)

The first is approximately 0.00483941449. The second is approximately 0.00483941449 at the displayed precision, but the two calculations are not mathematically identical. The approximation uses a flat rectangle while the cumulative difference integrates the curved density.

Use the cumulative difference for an interval probability. Wider intervals or other positions can make the approximation materially different. This example illustrates why density and probability have different meanings even when their numbers are related.

=PHI(2)/PHI(0)
=EXP(-2)

Both expected values are 0.135335283. The normalization factor cancels in the ratio. This compares how high the curve is at two positions; it does not compare the probabilities of two point events.

For tail work, avoid deriving probabilities by multiplying a density by an arbitrary width. Use a clearly defined interval and a cumulative distribution function instead.

Microsoft documents #VALUE! for invalid data types such as nonnumeric input and #NUM! for invalid numeric values. Negative finite inputs are ordinary points on the curve and should not be rejected merely for their sign.

If a centered input is expected to return 0.5, a cumulative function was probably intended. If a rescaled normal curve has the wrong vertical scale, check the division by standard deviation. If small tail values appear as zero on screen, inspect the number format before concluding that the underlying value is exactly zero.

Native Excel rounding, underflow limits, extreme inputs, blanks and text coercion were not tested. The formulas and numerical controls use an independent mathematical implementation, so they are reference expectations rather than captured spreadsheet output.

• Current documentation: the dedicated Microsoft reference lists Excel for Microsoft 365, Excel for Microsoft 365 for Mac, Excel 2024, Excel 2024 for Mac, Excel 2021, Excel 2021 for Mac, Excel 2019, Excel 2016.
• Introduced: Microsoft’s alphabetical function catalog marks PHI as introduced in Excel 2013. The exact first build and platform-specific introduction dates are not established here.
• Last functioning version: unknown; a current help listing is not a last-working-build test.
• Evidence checked: October 15, 2026. Official documentation was freshly retrieved; examples below were independently calculated. No native Excel, Mac or web-client execution was performed.

• Microsoft PHI reference.
• Microsoft function catalog and introduction-marker explanation.
• Editorial testing standards.

Examples use English function names and comma separators. Expected numbers are rounded independent mathematical calculations; documented error expectations are labeled as such. They are not captured Excel output. All illustrative data and policies are fictional.