Looking across Figure 24-4, you might have guessed that as N gets larger, the binomial distribution’s

shape approaches that of a normal distribution with mean

and standard deviation

.

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

with a standard deviation of

. Using this transformation, you can analyze data consisting

of observed proportions with t tests, ANOVAs, regression 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.

© John Wiley & Sons, Inc.

FIGURE 24-5: The Poisson distribution.

The formula to estimate probabilities on the Poisson distribution is

.

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

and standard deviation

.

The square roots of a set of Poisson-distributed numbers are approximately normally

distributed, with a standard deviation of 0.5.

The Exponential Distribution

If a set of events follows the Poisson distribution, the time intervals between consecutive events