# 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.

#### Table of Contents

## 1. Introduction

**What is a density function?**

A density function (PDF) 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.

A cumulative density function (CDF) represents the probability that the random variable will take a value less than or equal to a specified value.

**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 z-score?**

A z-score, also known as a standard score, is a measure of how many standard deviations a data point is away from the mean of a distribution.

The formula to calculate the z-score is:

z = (x - μ) / σ

x = the data point

μ = the mean or average of the distribution

σ = the standard deviation of the distribution

**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,FALSE) for a standard normal distribution. The NORM.DIST function can calculate both the cumulative and the dnesity functions, the third argument TRUE or FALSE determines the distribution function.

## 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 1

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

Cell range B3:B19 contains the x values starting with -4 with a step value of 0.5. Cell range C3:C19 contains the PHI function, it calculates the density function based on the corresponding x values in B3:B19.

Formula in cell C3:

The image above also shows a chart displaying the standard normal distribution using the PHI function to calculate the actual values.

## PHI Function Example 2

**A company measures the weight of its product packages. The weights are normally distributed with a mean of 500 grams and a standard deviation of 20 grams. What is the probability that a randomly selected package will weigh exactly 530 grams?**

The z-score is calculated like this:

z = (x - μ) / σ

x = the data point

μ = the mean or average of the distribution

σ = the standard deviation of the distribution

(C18-C19)/C20 becomes (530-500)/20 = 1.5 This value is calculated in cell C22.

Formula in cell C24:

The formula returns 0.12952 (12.95%) which means this is the probability of a package weighing 530 based on a mean of 500 and a standard deviation of 20. This is true if the observations follow a normal distribution.

The image above shows a graph displaying the normal distribution. A vertical line shows the x value of 1.5, and the horizontal line shows the y value of approx 0.1295. The intersection between these lines also shows the data point on the blue curve which represents the normal distribution.

## PHI Function Example 3

**A researcher measured an interesting process in nature, 40 observations were made, shown below. Calculate the mean, standard deviation. What is the probability that an observation is exactly 22? Here are the observations:
**

23 | 8 | 29 | 33 |

36 | 38 | 36 | 42 |

7 | 20 | 37 | 20 |

26 | 50 | 25 | 31 |

28 | 29 | 38 | 37 |

33 | 27 | 34 | 29 |

32 | 43 | 38 | 31 |

30 | 28 | 23 | 41 |

27 | 26 | 25 | 24 |

28 | 27 | 38 | 44 |

To calculate the z-score we first need to calculate the mean using the AVERAGE function. Cell C19 calculates the mean:

The calculation in cell C19 returns 30.5 based on the observations in cell range F19:I28. We also need the standard deviation to calculate the z-score. The STDEV.S function allows us to calculate the standard deviation based on a sample of the population. Cell C20 calculates the standard deviation:

To calculate the z-score:

z = (x - μ) / σ

x = the data point

μ = the mean or average of the distribution

σ = the standard deviation of the distribution

Cell C22 calculates the z-score based on the x value in C18 minus the the average in c19, the difference is then divided by the standard deviation in cell C20:

Cell C23 calculates the probability:

The formula returns 0.247 (24.7%) which means this is the probability of an observation is 22 based on a mean of 30.5 and a standard deviation of 8.7. This is true if the observations follow a normal distribution.

The image above shows a graph displaying the normal distribution. A vertical line shows the x value (z-score) of -0.979, and the horizontal line shows the y value of approx 0.25. The intersection between these lines also shows the data point on the blue curve which represents the normal distribution.

## 7. 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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