# How to use the NORM.DIST function

**What is the NORM.DIST function?**

The NORM.DIST function calculates the normal distribution for a given mean and standard deviation.

#### Table of Contents

## 1. Introduction

**What is a normal distribution?**

The normal distribution is a symmetric bell-shaped probability distribution described by its mean and standard deviation. Used by many to model a plethora of natural phenomena and represent unknown processes.

**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.

**What is deviation?**

In statistics, deviation is a measure of how far each value in a data set lies from the mean (the average of all values). A high deviation means that the values are spread out widely, while a low deviation means that they are clustered closely around the mean.

**What is standard deviation?**

Standard deviation measures dispersion from the mean by taking the square root of the average of squared deviations, useful for assessing variability and spread in data.

**What is the difference between deviation and standard deviation?**

**Deviation**is the difference between an individual data point and the mean.**Standard deviation**measures the variation across all deviations by using the square root of the average squared deviation.

**What is the difference between the NORM.DIST function vs NORM.INV function?**

NORM.INV function returns the inverse of the normal cumulative distribution for a given mean and standard deviation.

NORM.INV(*probability*, *mean*, *standard_dev*)

The NORM.DIST function calculates the normal distribution for a given mean and standard deviation.

NORM.DIST(*x, mean, standard_dev, cumulative)*

For example, the chart above demonstrates a normal distribution with a mean of 0 (zero) and standard deviation of 1.

The NORM.DIST(-1,0,1,TRUE) returns 0.158655253931457 which is the orange area below the curve up to x = -1 that represents the cumulative probability.

The NORM.INV(0.158655253931457,0,1) returns -1 which is the x value given the probability of 0.158655253931457

**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.

## 2. NORM.DIST function Syntax

NORM.DIST(*x*, *mean*, *standard_dev*, *cumulative*)

## 3. NORM.DIST function Arguments

x |
Required. A number to calculate the distribution for. |

mean |
Required. The average of the distribution. |

standard_dev |
Required. The standard deviation of the distribution. |

cumulative |
Required. A boolean value that determines which distribution the NORM.DIST function returns. TRUE - Cumulative distribution FALSE - Probability mass function |

If mean = 0, standard_dev = 1, and cumulative = TRUE, the standard normal distribution is returned.

If cumulative = TRUE, the formula is calculated the integral from negative infinity to x.

**What is the probability mass function?**

NORM.DIST with the cumulative parameter set to FALSE returns the value of the probability density function which is the value at the y-axis for a given x-axis value. The image above shows the y value 0.24197 for x-axis value -1.

NORM.DIST(-1,0,1,FALSE) returns approx. 0.24197

NORM.DIST(-1,0,1,TRUE) returns approx. .15866 which is the integral from negative infinity to -1

The image above shows the integral from negative infinity to x axis value -1.

## 4. NORM.DIST function Example 1

**In a manufacturing process, the diameter of a particular component follows a normal distribution with a mean of 10 mm and a standard deviation of 0.2 mm. What is the probability that a randomly selected component will have a diameter less than 9.8 mm?**

The arguments are:

- x = 9.8 mm
- mean = 10 mm
- standard_dev = 0.2
- Cumulative = True

These arguments are specified in cells C17,C18,C19,and C20 respectively, in the image above.

The image above shows a chart containing a blue curve representing the probability mass function of a normal distribution where the mean is 10 and the standard deviation is 0.2. The black curve represents the cumulative distribution based on the same mean and standard deviation as described above.

Formula in cell C22:

The formula returns 0.159 which represents the area below the blue curve between 0 (zero) and x= 9.8.

In the image above, locate the value 9.8 on the x-axis. From that point, draw an imaginary vertical line upwards until it intersects with the black curve, which represents the cumulative distribution function. Then, follow the point of intersection horizontally towards the y-axis to the right. You will find that the corresponding value on the y-axis is approximately 0.159.

## 5. NORM.DIST Function Example 2

**The daily returns of a particular stock follow a normal distribution with a mean of 0.05% and a standard deviation of 1.2%. What is the probability that the daily return will be greater than 2%?**

The arguments are:

- x = 2 %
- mean = 0.05 %
- standard_dev = 1.2%
- Cumulative = True

These arguments are specified in cells C17,C18,C19,and C20 respectively, in the image above.

The image above shows a chart containing a blue curve representing the probability mass function of a normal distribution where the mean is 0.05 and the standard deviation is 1.2. The black curve represents the cumulative distribution based on the same mean and standard deviation as described above.

Formula in cell C22:

The formula returns 0.052 (5.2%) which represents the area below the blue curve between x= 2 and infinity.

In the image above, locate the value 2 on the x-axis. From that point, draw an imaginary vertical line upwards until it intersects with the black curve, which represents the cumulative distribution function. Then, follow the point of intersection horizontally towards the y-axis to the right. You will find that the corresponding value on the y-axis is approximately 0.95. This value represents the area below the blue curve between x = 0 and x = 2, however, we need the area below the blue curve from x=2 to x = infinity. We need to calculate the complement to get the correct value.

## 6. NORM.DIST function not working

The NORM.DIST returns a

- #VALUE! error value if the
*mean*or*standard_dev*is nonnumeric. - #NUM! error value if
*standard_dev*≤ 0

## 7. How is the NORM.DIST function calculated?

The NORM.DIST function is very useful if you are working with statistics. Here is how the function works in detail (cumulative = FALSE).

*µ* is the mean.

*σ* is the standard deviation.

Use the AVERAGE function to calculate the arithmetic mean, used in the second argument in the NORM.DIST function.

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### Functions in 'Statistical' category

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

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