PHI function

The PHI function in Excel returns the standard normal distribution’s probability density function (PDF) for a given value. This function is typically used in statistics to calculate the likelihood or density of a specific value occurring within a standard normal distribution (mean = 0, standard deviation = 1).

Syntax:

PHI(z)

Arguments:

• z: Required. The z-score for which you want to calculate the probability density function. A z-score represents the number of standard deviations a value is away from the mean of the distribution.

Output:

The function returns the probability density corresponding to the given z-score under the standard normal curve.

Formula:

The formula for the standard normal distribution’s PDF is:

$$f(z) = \frac{1}{\sqrt{2\pi}} \cdot e^{-\frac{z^2}{2}}$$

Where:

• z is the z-score,
• e is Euler’s number (approximately 2.71828),
• π is the mathematical constant Pi (approximately 3.14159).

This formula calculates the probability density at a given z-score under the standard normal curve.

Example 1: Calculating the Probability Density for a Z-Score

Suppose you want to find the probability density for a z-score of 1.5.

Use the formula:

=PHI(1.5)

This will return the probability density corresponding to a z-score of 1.5 in the standard normal distribution.

Example 2: Calculating the Probability Density for a Negative Z-Score

If you want to find the probability density for a z-score of -2, use the formula:

=PHI(-2)

This will return the probability density corresponding to a z-score of -2 in the standard normal distribution.

Key Points:

• The PHI function returns the probability density for a given z-score, assuming the data follows a standard normal distribution (mean = 0, standard deviation = 1).
• The z-score is a measure of how many standard deviations a value is away from the mean of the distribution.
• The result gives the height of the standard normal curve at the specified z-score, not the cumulative probability.

Use Cases:

• Statistics: Determine the probability density of a value in a standard normal distribution.
• Risk Management: Understand the likelihood of a particular outcome occurring under a normal distribution.
• Data Analysis: Calculate the likelihood of a value occurring in standardized datasets.

Notes:

• PHI only works for the standard normal distribution (mean = 0, standard deviation = 1). If you need to work with normal distributions with different means or standard deviations, use the NORM.DIST function instead.
• The output from PHI(z) represents the height of the normal distribution curve, not the cumulative probability for that value.