iSixSigma

Poisson Distribution

Definition of Poisson Distribution:

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In statistics, distributions can be described as either continuous or discrete. The normal distribution is the most common of the continuous distributions. The binomial and Poisson distributions are common examples of discrete distributions. Let’s look at the characteristics and uses of the Poisson distribution.

Overview: What is the Poisson distribution? 

The Poisson distribution is named after the French mathematician Simeon Denis Poisson. The Poisson distribution is a discrete probability distribution that provides the probability of a given number of events occurring in a specified interval of time or space. The occurrence of these events should have a known constant mean rate (lambda) and be independent of the other events.

The Poisson distribution is appropriate if the following conditions are met:

  • X is the number of times an event occurs in a specified interval of time or space and X can take on any discrete values of zero and above.
  • Events are independent, meaning the occurrence of one event does not affect the probability of the next event. 
  • The average rate at which events occur is usually assumed to be constant.
  • Two events cannot occur at the same time.

An industry example of the Poisson distribution 

A call center collected data on its call volume. The Six Sigma Black Belt (BB) calculated there was an average of 300 incoming calls per hour, 24 hours a day. The calls are independent since an incoming call does not change the probability of when the next call will come in. The number of calls received during any minute has a Poisson probability distribution with a lambda or mean of 5 calls per minute (300 calls per hour/60 minutes/hour).

The BB wanted to determine the probability of 6, 8 or 10 calls coming in per minute so she could help plan for staffing levels. Using the Poisson distribution, the BB calculated the probability of 6, 8 and 10 calls per minute as follows:

Events      Probability

  6               0.23782

  8               0.06809

10               0.01370

A decision was made to plan for up to 8 calls per minute and not to worry about 10 since the probability of that volume was small. 

Frequently Asked Questions (FAQ) about the Poisson distribution

What are the three assumptions of the Poisson distribution? 

The three underlying assumptions or conditions of the Poisson distribution are:

  1. The occurrence of one event does not affect the probability of occurrence for another event. Events are independent. 
  2. The average rate is constant. The number of events per time period doesn’t change over time.
  3. Two events cannot occur simultaneously.

What is the difference between a binomial and Poisson distribution? 

There are only two outcomes for a binomial distribution. For example, this could be heads/tails or pass/fail. The Poisson has the possibility of an infinite number of events during a fixed period of time. Both are discrete distributions. 

The shape of the Poisson distribution is determined by what factor? 

Lambda is the mean number of events within a specified period of time. If lambda is small, the distribution will be positively skewed. The Poisson distribution will approach the shape of the normal distribution when lambda is approximately 20.

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