Probability Distributions

Fundamentals of Social Statistics by Adam J. McKee

We will consider how to use probability distributions in some detail in later sections, but for now, we want to get an idea of what the term means.  A probability distribution is a formula or a table used to assign probabilities to each possible value of a random variable X.  As with variables, a probability distribution can be discrete, which means that X can assume one of a finite (countable) number of values, or continuous, in which case X can assume one of an infinite number of different values. For the coin-flipping experiment in the above example, the probability distribution of X could be a simple table that shows the probability of each possible value of X, written as P(X).  The symbol P(X) is read “the probability of X” and is a shorthand way of saying it, especially in probability problems.

Assume that a random variable X is defined as the number of “heads” that turn up during the course of this experiment.  Coin flipping examples are used so often in probability that everyone familiar with the discipline knows that H stands for “heads,” and T stands for “tails.”   X assigns values to the outcomes of this experiment as follows:

Outcome X
{TT} 0
{HT, TH} 1
{HH} 2

Note that the “curly brackets” are used in set notation.  Each T in brackets indicates the results of a single flip, so the first result {TT} tells us that it is possible to get a result of “tails” for each of the two flips.  In that case, there would be zero heads.  Since one flip could come out one way and the second flip could come out the other, the {HT, TH} notation shows this, and the fact that either way it goes, there will be a result of 1 “head.”  Finally, both tosses could result in heads, in which case there would be two “head” results, as depicted by the notation {HH}.  Since there are four equally likely possible outcomes, the probability of each would be the same as the frequency, which would be only one out of the four possible results:  ¼, or .25 (25%).

X P(X)
0 0.25
1 0.50
2 0.25

The above table represents a probability distribution for two coin flips.  As we would expect, the most common result would be to have a result of one head and one tail.  This is because it can happen in two different ways out of the possible four (2/4 = .5).  Since we’ve defined X as the number of “heads” we have, the most probable outcome of a random experiment is X = 1.

In probability problems, the author will often specify a "fair coin" meaning that the result of "head" and "tail" are equally likely; apparently, statistics professors generally are in possession of weighted coins.

Probability distributions can be discrete or continuous, depending on the nature of the trial.  Researchers can use the binomial distribution to compute probabilities for a process where only one of two possible outcomes may occur on each trial.  In other words, when the outcome is “pass” or “fail,” the binomial distribution would be useful to the researcher in determining the probability that a particular person or group of people would pass or fail.

Of the continuous distributions, the most commonly used are the uniform distribution and the normal distribution.  In this text, the normal distribution is the most important by far, but keep in mind that there are others.

The uniform distribution is useful because it represents variables that are equally distributed over a given interval. For example, if the length of time until the next traffic accident occurs at a particularly busy intersection is equally likely to be any value between one and thirty days, then a researcher could use the uniform distribution to calculate probabilities for the time until the next accident occurs.

The normal distribution is perhaps the most useful probability distribution and is used in a diverse array of applications across many disciplines.  There some incredibly useful mathematical probabilities that are associated with the normal distribution, and we will use several of those to our advantage in later sections.  The normal distribution is characterized by probabilities (or scores) clustering around the mean or “center” of the distribution.  As you move farther and farther from the average, the less probable (frequently) an observation becomes.  In our everyday life, we observe that height is normally distributed.  If we see a man that is 6′ 1″ tall, we don’t think much of it.  If we see a man who is 7′ 3″ tall, we have to make an effort not to stare.  Our six-footer is a little tall, but close enough to the mean that we see people of that height on a fairly regular basis.  When we see a seven-footer, we take notice because we encounter men of that height infrequently.  The same thing can be said for hat sizes, ring sizes, shoe sizes, and about anything else that is found in nature.

The normal distribution is characterized by the classic bell-shaped curve, and when considering probability distributions, the areas under the curve represent (as you already guessed) probabilities.

The normal distribution has many useful statistical properties that make it a popular choice for statistical modeling (often, even when it shouldn’t be used). One of these properties is known as symmetry, the idea that the probabilities of values below the mean are matched by the probabilities of values that are equally far above the mean.  In other words, one half of the curve is the mirror image of the other half.

We will delve deeper into the normal curve later, and what we say about the normal curve in the context of frequency distributions is equally true for probability distributions.  The characteristics of the normal curve don’t change, regardless of what we are measuring.  At this stage, just keep in mind that probability distributions are useful because they let us determine the probability of a particular outcome of a particular experiment if the event actually follows the selected distribution.

Also, keep in mind that any curve that can be graphed can also be described by an equation.  Those equations can be fed into computer software such as Excel, such that when we “feed-in” a particular score that follows a particular probability distribution, the software can give us back the probability using the equation.  Of course, all of the formulas are hidden “under the hood,” and we just see a probability magically appear in a cell where we requested it.

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Last Modified:  06/03/2021

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