# Poisson & Bi-Nomial Distributions: What's the difference? When to use?

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#### Al Dyer

Can anyone give the down and dirty difference between bi-nomial and poisson distributions and when they can and should be used?

ASD...

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#### Atul Khandekar

The binomial distribution is useful for describing distributions of binomial events,
such as the number of males and females in a random sample of companies, or
the number of defective components in samples of 20 units taken from a production process.

The Poisson distribution is also sometimes referred to as the distribution of rare events.
Examples of Poisson distributed variables are number of accidents per person,
number of sweepstakes won per person, or the number of catastrophic defects found in a
production process.

For more details, equations etc. go to: https://www.statsoft.com/textbook/stathome.html

[This message has been edited by Atul Khandekar (edited 08 May 2001).]

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#### MarkR

I'll take a shot at this one. Binomial covers every situation, Poisson is an 'easier' approximation that is useful in certain situations. Because Binomial contains the n! term, if n > 69 (n is sample size) many calculators run out of steam trying to calculate such a huge number. Poisson gets around this by not using n!, just k!.

Poisson is appropriate when the sample is very large and the chance of any defect is small. You can prove this to yourself using Excel. Create 3 graphs, one using the binomial with n=200 and p=0.03, another using n=20 and p=0.3, and the third using Poisson np=6. (note that all three of these graphs have the same expected value, 6) If you plot all three on the same axis you will see that the B(200,0.03) curve and the Poisson curve are very similar, but the B(20, 0.3) curve is markedly different.

So which is correct? Binomial is always correct, these graphs just show that Poisson 'approximates' the binomial best when the conditions of large sample and small p are met.

Another useful application for Poisson is when np is known, but not n nor p. For example, a contractor averages 7 new homes every year (np = 7). What is the number of people that could call the contractor ? It's huge, but we don't know what it is. What is the probability that any one person in the population that might call wants a house built that year(p)? We don't know, but probably small. But we do know np=7, so we could calculate the probability of the contractor building 6 houses next year, or 8 houses next year.

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#### Rick Goodson

Al,

As you requested, down and dirty:

The binomial probability distribution is applicable to discrete problems involving an infinite number of items or a steady stream of items from a work center. It applies to problems where there are attributes such as good/bad, success/failure, yes/no, etc.

The Poisson probability distribution is applicable to problems involving an observation per unit of time such as cars through a toll gate or number of defects in a 1000 square yards of fabric. The Poisson is applicable when the sample size is quite large and the average probability of an event or defect (p') is quite small.

The Poisson distribution is easier to calculate than the binomial and can be used as an approximation to the binomial when p' is equal to or less than 0.10 and np' is equal to or less than 5.

Hope that helps.

Rick

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#### Ken K.

Check out
https://www.itl.nist.gov/div898/handbook/eda/section3/eda366.htm

The binomial distribution is usually used for count data and proportion data that involves two states of existance (success or failure). Examples: Number of defects, propportion of satisfied customers, etc..

The poisson is usually used when you are looking at the number of "things" (defects, whatever) observed over some quasi-continuous quanity of sample. Examples: waiting time for defects, defects per xxx square feet of aluminum foil, bubbles per cubic feet of lexan, etc...

Data that involve the number of successes (x) out of some number of trials , where the probability of a success, p=x/n, is fixed, is said to follow a Binomial{n,p} distribution. Note that this is a discrete distribution, since x consists of integers falling between 0 and n, inclusive. All proportions follow a binomial distribution. As p becomes small and n becomes large (p<0.01, n>100), the binomial distribution tends to have a nearly continuous, but skewed shape, quite similar to the shape of the Poisson{lambda=p} distribution. As p continues to get smaller and n continues to get larger (np> 15), the binomial distribution appears even more continuous and becomes more symmetric about the mean, approaching the shape of the Normal{mu=p, sigma^2=2p(1-p)/n} distribution.

The word "success" here can mean any condition - it can even mean a defect. In the same way, "failure" just means the other condition.

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