# How to use the PHI function

**What is the PHI function?**

The PHI function calculates a number of the density function for a standard normal distribution.

**What is a density function?**

A density function in statistics describes the relative likelihood that a random variable takes on a given value. The area under the entire density function integrates to 1.

There are two main types of density functions:

- Probability Density Function (PDF)

PDFs are used for continuous random variables and provides the probability that the variable's value lies within a small range around x.

The area under PDF over an interval provides the probability that the variable lies within that interval. - Probability Mass Function (PMF)

PMFs are used for discrete random variables and gives the exact probability that a variable takes on a specific value x.

**What is the difference between continuous random variables and discrete random variables?**

Continuous random variables can take on any value within a continuous range. Examples: height, weight, temperature.

Discrete random variables can only take on specific discrete values within a finite/countable set. Examples: number of items, dice rolls.

**What is a normal distribution?**

A normal distribution, also known as a Gaussian distribution, is a continuous probability distribution that is symmetric about the mean and defined by two parameters - the mean (μ) and standard deviation (σ).

The formula for the normal probability density function is:

f(x) = (1 / (σ√(2π))) * e^((-1/2) * ((x - μ) / σ)^2)

Where:

f(x) = probability density function

x = value

μ = population mean

σ = population standard deviation

e = exponential constant (Euler's number)

π = mathematical constant pi

**What is a standard normal distribution?**

A standard normal distribution is a normal distribution with the mean of 0 (zero) and the standard deviation is 1. You can standardize any normal distribution using the STANDARDIZE function in Excel, it works like this:

z = (x - *µ)/σ*

*z = z-score
µ* is the mean.

*σ*is the standard deviation.

**What is the mean?**

It is also known as the average. It is calculated by adding up all the values in the data set and dividing by the number of values.

For example, if you have a data set of 5, 7, 9, 11, and 13, the mean is (5 + 7 + 9 + 11 + 13) / 5 = 9.

Use the AVERAGE function to calculate the mean.

**What is standard deviation?**

Standard deviation is a measure of dispersion that indicates how spread out the values in a dataset are from the mean. It is represented by the Greek letter sigma (σ).

The formula for calculating standard deviation is:

σ = √Σ(x - μ)2 / (N - 1)

Where:

σ = Standard deviation

Σ = Sum of

x = Values in the dataset

μ = Mean of the dataset

N = Number of values in the dataset

(N - 1) = Sample correction factor

Use the STDEV.S function or STDEV.P function to calculate the standard deviation.

**What is the difference between the PHI function and the NORM.DIST function?**

PHI has no arguments for mean and standard deviation. It assumes a standard normal distribution (mean 0, standard deviation 1). NORM.DIST allows specifying mean and standard deviation.

PHI(x) is equivalent to NORM.DIST(x,0,1,TRUE) for a standard normal distribution.

### PHI function Syntax

PHI(*x*)

### PHI function Arguments

x |
Required. x is the number for which you want to calculate the density of the standard normal distribution. |

### PHI function example

This example demonstrates the different values of the standard normal distribution between x=-4 and x=4 with an increment of 0.5

Formula in cell C3:

The line chart demonstrates the standard normal distribution based on values from the PHI function.

### PHI function not working

The PHI function returns:

- #NUM# error value if the argument is an invalid numeric value.
- #VALUE! error value if the argument is an invalid data type, like a non numeric value.

### Functions in 'Statistical' category

The PHI function function is one of 73 functions in the 'Statistical' category.

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