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When we speak of a mathematical variable (or simply, variable), we mean a symbol that at any point could be replaced by values contained in a specified set. For instance, consider the mathematical variable y_i. By the subscript i is indicated the fact that y_i stands for a set of values, not all equal to the same number (otherwise y would be a constant) such that at any point in time any of these values in the set could serve as a temporary “replacement” for the symbol.

Of course, social and natural sciences are all about variables. Here are some examples:

 Height of persons in the world is a variable because persons of the world have different heights. However, height would be considered a constant if 10 people in a room were of the exact same height (and those were the only people we were considering).

 Blood pressure is a variable because persons, animals, and other living creatures have different blood pressure measurements.

 Intelligence (IQ) of human beings (difficult to measure to be sure, though psychology has developed instruments in an attempt to assess such things) is a variable because presumably people have differing intellectual capacities.

 Earned run average (ERA) of baseball players is a variable because players do not all have the same ERA.

A random variable is a mathematical variable that is associated with a probability distribution. That is, as soon as we assign probabilities to values of the variable, we have a random variable. More formally, we can say that a random variable is a function from a sample space into the real numbers (Casella and Berger, 2002), which essentially means that elements in the set (i.e., sample space) have probabilities associated with them (Dowdy, Wearden, and Chilko, 2004).

Consider a simple comparison between a mathematical variable and a discrete random variable in Table 2.4.

Notice that for the mathematical variable, probability does not enter the picture, it is not of any consideration. For the discrete random variable, each value of the variable has a probability associated with it. Note as well that the probabilities must sum to 1.0 for it to be a legitimate probability distribution (i.e., 0.20 + 0.50 + 0.30 = 1.0). How the given probabilities are assigned is a matter to be governed by the specific context of the problem. Recall as well that variables can be classified as discrete or continuous (see Appendix for a review). This same distinction can be applied to random variables as to ordinary mathematical variables. In Table 2.4 features a discrete random variable. For continuous random variables, since the probability of any particular value in a continuous distribution is theoretically zero, instead of associating probabilities with particular values, probabilities are associated with areas under the curve computed by way of integration in calculus.

Table 2.4 Mathematical versus Discrete Random Variable

Mathematical Variable yi	Random Variable yi
y 1 = 1	y 1 = 1 (p = 0.20)
y 2 = 3	y 2 = 3 (p = 0.50)
y 3 = 5	y 3 = 5 (p = 0.30)

The distinction between mathematical and random variables is important when we discuss such things as means, variances, and covariances. A reader first learning about random variables, having already mastered the concept of sample or population variance (to be discussed shortly), can be somewhat taken aback when encountering the variance of a random variable, given as

and then attempting to compare it to the more familiar variance of a population:

Realize, however, that both expressions are essentially similar, they both account for squared deviations from the mean. However, the variance of a random variable is stated in terms of its expectation, E. Throughout this book, we will see the operator E at work. What is an expectation? The expectation E of a random variable is the mean of that random variable, which amounts to it being a probability‐weighted average (Gill, 2006). The operator E occurs several times throughout this book because in theoretical statistics, long‐run averages of a statistic are of especial interest. As noted by Feller (1968, p. 221), should an experiment be repeated n times under identical conditions, the average of such trials should be close to expectation. Perhaps less formally, the operator E then tells us what we might expect to see in the long run for large n. Theoretical statisticians love taking expectations, because the short run of a variable is seldom of interest at a theoretical level. It is the long (probability) run that is often of most theoretical interest. As a crude analogy, on a personal level, you may be “up” or “down” now, but if your expectation E pointed to a favorable long‐run endpoint, then perhaps that is enough to convince you that though “on the way” is a rough tumbly road, in the end, as the spiritual would say, we “arrive” at our expectation (which perhaps some would denote as an afterlife of sorts).

The key point is that when we are working with expectations, we are working with probabilities. Thus, instead of summing squared deviations of the kind as one does in the sample or population variance for which there is specified n, one must rather assign to these squared deviations probabilities, which is what is essentially being communicated by the notation “E(y_i − μ)².” We can “unpack” this expression to read

where p(y_i) is the probability of the given deviation, (y_i − μ), for in this case, a discrete random variable.