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 many functions in the 'Statistical' category.

Excel function categories

Excel categories

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