# The Excel POISSON Function

Poisson Distribution

The Poisson distribution is a discrete probability function that is used to calculate the probability of a number of events occurring in a specified time period.

The Poisson probability mass function calculates the probability that there will be exactly x occurrences within the specified time period. This is given by the formula: where λ is the expected number of occurrences within the specified time period.

The cumulative Poisson probability function calculates the probability that there will be at most x occurrences within the specified time period. This is given by the formula: For further information, see the Wikipedia Poisson Distribution Page
POISSON and POISSON.DIST

In Excel 2010, the Poisson function has been replaced by the Poisson.Dist function, which has improved accuracy.

Although it has been replaced, the Poisson function is still available in Excel 2010 (stored in the list of compatibility functions), to allow compatibility with earlier versions of Excel.

## Function Description

The Excel POISSON function calculates the Poisson Probability Mass Function or the Cumulative Poisson Probability Function for a supplied set of parameters.

The syntax of the function is:

POISSON( x, mean, cumulative )

Where the function arguments are:

x-

The number of events that you want the probability for (must be ≥ 0).

This value should be an integer; If a decimal is supplied, it will be truncated to an integer by Excel.
mean-The expected number of events (must be ≥ 0).
cumulative-

A logical argument that specifies the type of distribution to be calculated.

This can be either:

 TRUE - Use the cumulative distribution function FALSE - Use the probability mass function

## Poisson Function Examples

### Example 1 - Poisson Probability Mass Function Poisson Probability Mass Function with mean (λ) = 25

The above chart on the right shows the Poisson Probability Mass Function with an expected value (mean) of 25.

If you want to use Excel to calculate the value of this function at x = 20, this can be done using the Excel Poisson function, as follows:

=POISSON( 20, 25, FALSE )

This gives the result 0.051917469.

### Example 2 - Cumulative Poisson Probability Function Cumulative Poisson Probability Function with mean (λ) = 40

The above chart on the right shows the Cumulative Poisson Probability Function with an expected value (mean) of 40.

If you want to use Excel to calculate the value of this function at x = 35, this can be done using the Excel Poisson function, as follows:

=POISSON( 35, 40, TRUE )

This gives the result 0.242414198.

For further information and examples of the Excel Poisson function, see the Microsoft Office website.

## Poisson Function Errors

If you get an error from the Excel Poisson function this is likely to be one of the following:

Common Errors
 #NUM! - Occurs if either the supplied value of x, or the supplied mean is < 0. #VALUE! - Occurs if any of the supplied arguments are non-numeric.