# How to use the LOGNORM.DIST function

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

The LOGNORMDIST function calculates the probabiltity based on the lognormal distribution of argument x, mean, and std_dev. This function has replaced the LOGNORMDIST function.

## 1. Introduction

**What is the lognormal distribution?**

The lognormal distribution is a continuous probability distribution of a random variable whose logarithm follows a normal distribution. It is often used to model a steady relative growth rate is steady like financial returns.

**What is the normally distributed ln(x)?**

For a random variable x that follows a lognormal distribution like ln(x) is normally distributed. This means that if you take the natural logarithm of x the result will follow a standard normal distribution.

**Wha****t is a ****continuous probability distribution?**

A continuous probability distribution is defined over an interval and range of continuous values. This gives the probability an outcome that is exactly equal to any value, and having an area under its probability density curve equal to 1.

**What is a continuous value?**

A continuous value comes from a continuum of possible points rather than distinct separate values. It is able to take on any quantity within an interval rather than certain fixed outcomes.

## 2. LOGNORMDIST Function Syntax

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

## 3. LOGNORMDIST Function Arguments

x |
Required. |

mean |
Required. A value representing the mean of ln(x). |

standard_dev |
Required. A value representing the standard deviation of ln(x). |

cumulative |
Required. A boolean value determining the form of the function. TRUE - cumulative distribution function FALSE - probability density function |

**What is the mean?**

The arithmetic mean is calculated by dividing the sum of all values by the number of values.

For example, an array contains these values: 3,2,1

The sum is 3 + 2 + 1 equals 6

The number of values is 3.

6/3 equals 2. The average of 3, 2, 1 is 2

**What is the standard deviation?**

Standard deviation tells you how far from the average values are spread out. Both charts above have numbers and an average plotted, they share the same average however, the numbers are not the same.

Chart A above shows that the values are more spread out than the values in chart B. Chart A has a standard deviation of 23.45256334, standard deviation for chart B is 5.207075606. Standard deviation is used in statistics.

**What is the cumulative distribution function?**

The cumulative distribution function defines the probability that a random variable is less than or equal to a specified value. It gives the area under the probability density curve up to that value.

**What is the probability density function?**

A probability density function defines a continuous probability distribution by providing the relative likelihood that a random variable takes on different values. The total area under the curve over all values equal to 1.

## 4. LOGNORMDIST Function Example 1

**In a manufacturing process, the diameter of a particular component follows a lognormal distribution with a mean logarithm of 2.1 and a standard deviation of the logarithms of 0.3. What is the probability that a randomly selected component will have a diameter greater than 10 units?**

The LOGNORM.DIST arguments are:

- x = 10
- mean = 2.1
- standard_dev = 0.3
- cumulative = true

The formula is LOGNORM.DIST(C16,C17,C18,C19) however we need the cumulative value for diameters above 10 units, not equal to and below 10 units. We need to calculate the complement by subtracting 1 with LOGNORM.DIST(C16,C17,C18,C19).

Formula in cell C7:

LOGNORM.DIST(C16,C17,C18,C19) returns approx. 0.75, 1 - LOGNORM.DIST(C16,C17,C18,C19) equals approx 0.25

The chart above shows blue columns representing the probability density function of the lognormal distribution and the black line represents the cumulative probability function.

In the image above, locate the value 10 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 0.75.

## 5. LOGNORMDIST Function Example 2

**The concentration of a certain pollutant in a river follows a lognormal distribution with a mean logarithm of 1.8 and a standard deviation of the logarithms of 0.2. What is the probability that a randomly sampled water sample will have a pollutant concentration exceeding the safe limit of 8 parts per million?**

The LOGNORM.DIST arguments are:

- x = 8
- mean = 1.8
- standard_dev = 0.2
- cumulative = true

The formula is LOGNORM.DIST(C16,C17,C18,C19) however we need the cumulative value for concentrations above 8 ppm, not equal to and below 8 ppm. We need to calculate the complement by subtracting 1 with LOGNORM.DIST(C16,C17,C18,C19).

Formula in cell C7:

LOGNORM.DIST(C16,C17,C18,C19) returns approx. 0.92, 1 - LOGNORM.DIST(C16,C17,C18,C19) equals approx 0.08

The chart above shows blue columns representing the probability density function of the lognormal distribution and the black line represents the cumulative probability function.

In the image above, locate the value 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 0.92.

## 6. LOGNORMDIST Function Example 3

**The lifespans of a certain species of insect follow a lognormal distribution with a mean logarithm of 1.5 and a standard deviation of the logarithms of 0.3. What is the probability that a randomly selected insect will live for less than 6 weeks?**

The LOGNORM.DIST arguments are:

- x = 6
- mean = 1.5
- standard_dev = 0.3
- cumulative = true

The formula is LOGNORM.DIST(C16,C17,C18,C19) however we need the cumulative value for the lifespan below 6 weeks.

Formula in cell C7:

LOGNORM.DIST(C16,C17,C18,C19) returns approx. 0.835

The chart above shows blue columns representing the probability density function of the lognormal distribution and the black line represents the cumulative probability function.

In the image above, locate the value 6 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 0.83.

## 7. LOGNORMDIST Function not working

The LOGNORM.DIST function returns

- #VALUE! error value if any argument is non-numeric.
- #NUM! error value if:
- x <= 0
- standard_dev <= 0

## 8. How is the LOGNORMDIST Function calculated

The equation to calculate the lognormal cumulative distribution is:

LOGNORM.DIST(*x,µ,σ*) = NORM.S.DIST(*ln(x)-µ / σ*)

### Functions in 'Statistical' category

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

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