Chapter 24

Ten Distributions Worth Knowing

IN THIS CHAPTER

Delving into distributions that may describe your data

Digging into distributions that arise during statistical significance testing

This chapter describes ten statistical distribution functions you’ll probably encounter in biological

research. For each one, we provide a graph of what that distribution looks like, as well as some useful

or interesting facts and formulas. You find two general types of distributions here:

Distributions that describe random fluctuations in observed data: Your study data will often

conform to one of the first seven common distributions. In general, these distributions have one or

two adjustable parameters that allow them to fit the fluctuations in your observed data.

Common test statistic distributions: The last three distributions don’t describe your observed

data. Instead, they describe how a test statistic that is calculated as part of a statistical significance

test will fluctuate if the null hypothesis is true. The Student t, chi-square, and Fisher F distributions

allow you to calculate test statistics to help you decide if observed differences between groups,

associations between variables, and other effects you want to test should be interpreted as due to

random fluctuations or not. If the apparent effects in your data are due only to random fluctuations,

then you will fail to reject the null hypothesis. These distributions are used with the test statistics

to obtain p values, which indicate the statistical significance of the apparent effects. (See Chapter

3 for more information on significance testing and p values.)

This chapter provides a very short table of critical values for the t, chi-square, and F

distributions. A critical value is the value that your calculated test statistic must exceed in order

for you to declare statistical significance at the α = 0.05 level. For example, the critical value for

the normal distribution is 1.96 at α = 0.05.

The Uniform Distribution

The uniform distribution is the simplest distribution. It’s a continuous number between 0 and 1. To

generalize, it is a continuous number between a and b, with all values within that range equally likely

(see Figure 24-1). The uniform distribution has a mean value of

and a standard deviation of

. The uniform distribution arises in the following contexts:

Round-off errors are uniformly distributed. For example, a weight recorded as 85 kilograms (kg)

can be thought of as a uniformly distributed random variable between a = 84.5 kg and b = 85.5 kg.

This causes the mean to be (84.5 + 85.5)/2 = 85 kg, with a standard error of (84.4 – 84.5)/√12,