Objectives

This chapter describes the means by which we label and treat known and unknown values. Basically there are two types of observable data, and the abstract terminology for yet-to-be observed values should also reflect this distinction. We first talk here about the levels of measurement for observed values where the primary distinction is discrete versus continuous. We will then see that the probability functions used to describe the distribution of such variables preserves this distinction. Many of the topics here lead to the use of statistical analysis in the social sciences.

Levels of Measurement

It is important to classify data by the precision of measurement. Usually in the social sciences this is an inflexible condition because many times we must take data “as is” from some collecting source. The key distinction is between discrete data, which take on a set of categorical values, and continuous data, which take on values over the real number line (or some bounded subset of it). The difference can be subtle. While discreteness requires countability, it can be infinitely countable, such as the set of positive integers. In contrast, a continuous random variable takes on uncountably infinite values, even if only in some range of the real number line, like [0 :1], because any interval of the real line, finitely bounded or otherwise, contains an infinite number of rational and irrational numbers.

To see why this is an uncountably infinite set, consider any two points on the real number line. It is always possible to find a third point between them. Now consider finding a point that lies between the first point and this new point; another easy task.