### Nothing fishy about Poisson distributions as used by law departments

Three times I have referred to a statistical function called a Poisson distribution, yet I have never explained the actual computation (See my post of Jan. 20, 2006: one of many kinds of distributions of numbers; Aug. 16, 2006: predicts likelihood of event during a given time period; and June 15, 2009: relation to queuing theory.). Nor did I mention that it is important to understand that a Poisson distribution implies randomness in the underlying events.

Here is what I learned from StatTrek http://stattrek.com/Lesson2/Poisson.aspx. I will apply it to a hypothetical, EEOC charges filed against your company each quarter. Let’s say over the past few years on average the company has prevailed in 6 of them per quarter. Further, assume that dismissal of the charge is a success and anything else is not a success and that you want to know the likelihood that in the coming quarter you will succeed on 7 charges. (Perhaps your performance bonus depends on that?)

The forbidding equation for a Poisson distribution to calculate a probability is P(x; μ) = (e-μ) (μx) / x!. In the EEOC scenario described above you would read it as “The probability that exactly 7 charges are dismissed during the next quarter where the average has been 6 per quarter is equal to 2.71828 (e is the base of the natural logarithm system, and if that is unclear to you, ignore the explanation but use the approximate value, which is raised to the negative power of 6), multiplied by 6 raised to the 7th power (the average number of dismissals per quarter multiplied by itself seven times) divided by 7 factorial (7 times 6 times 5 times 4 times 3 times 2).

The handy calculator on the StatTrek tells me that the probability is 13.8 percent that you will prevail on precisely 7 EEOC charges. You can also find out various cumulative probabilities. For example, the probability on these facts that you will prevail next quarter on more than 7 EEOC charges is 25.6 percent.