LOGNORM.DIST

Calculates the probability density or cumulative distribution function of a lognormal distribution.

Syntax:

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

x is required, and the value that you want to use to calculate the lognormal distribution of.

mean is required, and is the mean of the natural logarithm of(ln) of x.

standard_dev is required, and is the standard deviation of the natural logarithm of x.

cumulative is required, and is a logical value:

TRUE: Returns the cumulative distribution function (the probability that a random variable is less than or equal to x).

FALSE: Returns the probability density function (the height of the lognormal distribution curve at x).

Example:

If x contains 120, mean contains 4.5, standard_dev contains 0.8 and cumulative contains TRUE:

LOGNORM.DIST(120, 4.5, 0.8, TRUE)

returns 0.640338853

x:

mean:

standard_dev:

Cumulative:

Result:

0.640338853

Application:

An application of the LOGNORM.DIST function is analyzing the distribution of wealth in a population. Wealth distribution often follows a lognormal distribution, as a few individuals hold a disproportionately large amount of wealth, while the majority have much less.

Here's an example using the LOGNORM.DIST function:

Scenario: We want to determine the probability of a randomly selected person having a net worth of less than or equal to a certain amount. We have a population whose net worth is modeled by a lognormal distribution with a mean of the natural logarithm of wealth (μ) of 11.5 and a standard deviation of the natural logarithm of wealth (σ) of 1.2.

Data Table:

	Net Worth (x)	Probability (P(X≤x)) Calculated using LOGNORM.DIST	Result for LOGNORM.DIST	Interpretation
	A	B	C	D
1	$10,000.00	LOGNORM.DIST(10000, 11.5, 1.2, TRUE)	0.0282	The probability of a person having a net worth of $10,000 or less is approximately 2.82%.
2	$50,000.00	LOGNORM.DIST(50000, 11.5, 1.2, TRUE)	0.2854	The probability of a person having a net worth of $50,000 or less is approximately 28.54%.
3	$1,000,000.00	LOGNORM.DIST(1000000, 11.5, 1.2, TRUE)	0.9732	The probability of a person having a net worth of $1,000,000 or less is approximately 97.32%.

Explanation:

x (Net Worth): The value for which we want to find the cumulative probability. In our table, these are $10,000, $50,000, and $1,000,000.
mean (μ): This is the mean of the natural logarithm of the distribution. In this case, it's 11.5. This value is derived from the characteristics of our lognormal distribution.
standard_dev (σ): This is the standard deviation of the natural logarithm of the distribution. In our case, it's 1.2.
cumulative (TRUE/FALSE):
- TRUE: This calculates the cumulative distribution function (CDF), which gives the probability that a random variable will be less than or equal to a certain value. This is what we use in our example to find the probability of a person having a net worth less than or equal to the specified amount.
- FALSE: This calculates the probability density function (PDF), which gives the probability density at a specific point. This is useful for plotting the shape of the distribution, but for finding cumulative probabilities, TRUE is the correct argument.

By entering the formulas from the table into a spreadsheet, you can get the actual probabilities, demonstrating how the LOGNORM.DIST function helps us understand the concentration and spread of wealth in a population. For instance, the result for $1,000,000 would show the percentage of the population with a net worth below that threshold.