How to Use the GAUSS and PHI Functions in Google Sheets to Calculate Standard Normal Probabilities

When conducting statistical modelling or risk assessment in Google Sheets, analysts frequently work with Gaussian distributions (the standard bell curve). While functions such as NORM.DIST and NORM.S.DIST provide comprehensive options for arbitrary means and standard deviations, Google Sheets also provides two streamlined, dedicated functions tailored specifically for standard normal distributions: GAUSS and PHI.

The GAUSS function calculates the probability that a random variable falls strictly between the distribution mean (zero) and a given number of standard deviations. Meanwhile, PHI evaluates the probability density function (PDF) value at a specific coordinate. In this guide, we explore the exact mathematical definitions, syntax, practical use cases, and comparisons with related statistical functions.

Mathematical Foundations: GAUSS vs PHI

Distinguishing between the cumulative area calculated by GAUSS and the point density returned by PHI is critical for accurate analysis:

• GAUSS(z): Calculates the area under the standard normal curve between 0 and z. Specifically, it represents NORM.S.DIST(z, TRUE) - 0.5. Because a standard normal curve is symmetrical with half its total area (0.5) falling to the left of zero, GAUSS(z) measures the one-sided probability offset from the mean.
• PHI(z): Evaluates the standard normal probability density function: f(z) = (1 / SQRT(2 * PI())) * EXP(-0.5 * z^2). It returns the height of the bell curve at coordinate z rather than an accumulated area.

Syntax and Parameter Constraints

Both functions accept a single real numerical input or cell reference:

=GAUSS(z)
=PHI(z)

Keep these technical details in mind when deploying them:

• Domain Range: Both functions accept any real numerical value from negative infinity to positive infinity.
• Symmetry Property: GAUSS(-z) = -GAUSS(z). If you evaluate a negative z-score, GAUSS returns a negative probability representing the area to the left of the mean. Conversely, PHI(z) is strictly positive and perfectly symmetrical: PHI(-z) = PHI(z).
• Non-Numeric Arguments: Referencing non-numeric text strings returns a #VALUE! error.

Comparative Reference Matrix

The table below illustrates benchmark outputs across common standard deviation intervals:

z-score	GAUSS(z) [Area between 0 and z]	PHI(z) [Curve Height at z]	NORM.S.DIST(z, TRUE) [Cumulative Area]
-2.0	-0.4772	0.0540	0.0228
-1.0	-0.3413	0.2420	0.1587
0.0	0.0000	0.3989	0.5000
+1.0	0.3413	0.2420	0.8413
+2.0	0.4772	0.0540	0.9772
+3.0	0.4987	0.0044	0.9987

Practical Implementation Examples

1. Calculating the Empirical Rule (68-95-99.7 Rule)

To verify the proportion of a population within ±1 standard deviation of the mean using GAUSS:

=GAUSS(1) * 2

This outputs approximately 0.6827 (or 68.27%). Multiplying by two accounts for both the positive and negative halves of the symmetrical distribution.

2. Plotting a Normal Bell Curve

To draw a smooth bell curve in Google Sheets:

1. In column A (cells A2:A82), generate z-scores from -4.0 to +4.0 in increments of 0.1 using =SEQUENCE(81, 1, -40, 1) / 10.
2. In cell B2, calculate the probability density using PHI:

   =ARRAYFORMULA(PHI(A2:A82))

3. Highlight columns A and B, navigate to Insert > Chart, and select a Smooth line chart to visualize the distribution.

Frequently Asked Questions

Why would I use GAUSS instead of NORM.S.DIST?

GAUSS is concise when your analytical objective requires measuring the probability mass between the center of a process and an observed deviation threshold, common in Six Sigma quality control and engineering tolerance testing.

Can PHI be used with non-standard normal distributions?

PHI operates strictly on standardized z-scores. To evaluate a normal distribution with an arbitrary mean (μ) and standard deviation (σ), convert your x-value first: =PHI((x - mean) / stdev) / stdev, or use NORM.DIST(x, mean, stdev, FALSE).