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Part III: Distributions and the Central Limit Theorem

In math you have variables like X and Y that take on certain values depending
on the problem (for example, the width of a rectangle), but in statistics the
variables change in a random way. By random, statisticians mean that you
don’t know exactly what the next outcome will be but you do know that cer-
tain outcomes happen more frequently than others; everything’s not 50-50.
(Like when I try to shoot baskets; it’s definitely not a 50% chance I’ll make one
and 50% chance I’ll miss. It’s more like 5% chance of making it and a 95%
chance of missing it.) You can use that information to better study data and
populations and make good decisions. (For example, don’t put me in your bas-
ketball game to shoot free throws.)
Data have different types: categorical and numerical (see Chapter 4). While
both types of data are associated with random variables, I discuss only
numerical random variables here (this falls in line with most intro stat
courses as well). For information on analyzing categorical variables, see
Chapters 6 and 19.
Discrete versus continuous
Numerical random variables represent counts and measurements. They
come in two different flavors: discrete and continuous, depending on the
type of outcomes that are possible.
✓ Discrete random variables: If the possible outcomes of a random vari-
able can be listed out using whole numbers (for example, 0, 1, 2 . . . , 10;
or 0, 1, 2, 3), the random variable is discrete.
✓ Continuous random variables: If the possible outcomes of a random
variable can only be described using an interval of real numbers (for
example, all real numbers from zero to infinity), the random variable
is continuous.
Discrete random variables typically represent counts — for example, the
number of people who voted yes for a smoking ban out of a random sample of
100 people (possible values are 0, 1, 2, . . . , 100); or the number of accidents at
a certain intersection over one year’s time (possible values are 0, 1, 2, . . .).
Discrete random variables have two classes: finite and countably infinite. A
discrete random variable is finite if its list of possible values has a fixed (finite)
number of elements in it (for example, the number of smoking ban support-
ers in a random sample of 100 voters has to be between 0 and 100). One very
common finite random variable is the binomial, which is discussed in this
chapter in detail.

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