CHAPTER 24 Ten Distributions Worth Knowing 355

The arc-sine of the square root of a set of proportions is approximately normally

distributed, with a standard deviation of 1

4

/

N . Using this transformation, you

can analyze data consisting of observed proportions with t tests, ANOVAs, regres-

sion models, and other methods designed for normally distributed data. For

example, using this transformation, you could use these methods to statistically

compare proportions of participants who responded to treatment in two different

treatment groups in a study. However, whenever you transform your data, it can

be challenging to back-transform the results and interpret them.

The Poisson Distribution

The Poisson distribution gives the probability of observing exactly N independent

random events in some interval of time or region of space if the mean event rate

is m. The Poisson distribution describes fluctuations of random event occurrences

seen in biology, such as the number of nuclear decay counts per minute, or the

number of pollen grains per square centimeter on a microscope slide. Figure 24-5

shows the Poisson distribution for three different values of m.

The

formula

to

estimate

probabilities

on

the

Poisson

distribution

is Pr

,

/

(N m

m e

N

N

m

)

.

Looking across Figure 24-5, you might have guessed that as m gets larger, the

Poisson distribution’s shape approaches that of a normal distribution, with

mean m and standard deviation

m.

The square roots of a set of Poisson-distributed numbers are approximately nor-

mally distributed, with a standard deviation of 0.5.

FIGURE 24-5:

The Poisson

distribution.

© John Wiley & Sons, Inc.