1.3.3 Interval and Ratio Variables

Going back to the original Do It! 1.2. exercise, I am sure that you found imagining the categories of exam test scores and age the easiest, as they would be simply numbers. Perhaps something like 30, 65, 72, 88, 95, etc.… points out of 100 in the former case (though I know you don’t want to imagine a test score of 30 on any exam!), and, if we’re imagining college students, something like 18, 19, 20, 22, 23, 24, etc…. years in the latter. Notice the major difference from the categories of the nominal and ordinal variables we discussed above: now we are working with numbers. Not only are the exam scores and age categories comprised of numbers (as opposed to words) but they are also ordered in measurable “distances”. In other words, there is a stable/unchangeable unit by which the “distance” between any two categories can be measured: a point in the exam scores case and a year in the age case. This unit is called unit of measurement for the interval and ratio variables.

Wait a second, you’re probably thinking now — the exam scores above lists 30, 65, 72… as categories, and a quick calculation reveals that the “distance” between 30 and 65 is thirty-five points, while the “distance” between 65 and 72 is only seven points. Thirty-five is clearly much bigger (five times bigger to be precise) than seven: isn’t that as arbitrary as the “distances” across the educational attainment categories above? Well, no. The difference is that for interval and ratio variables the information contained in the categories and their “distances” from each other is not simply of the more/less, bigger/smaller, left/right, etc. kind but is readily quantifiable and measurable in precise, stable units. In practical terms, you can specify exactly how much smaller/bigger a category is than another (i.e., 65 points is thirty-five points more than 30 points; a 22 years-old is two years older than a 20 year-old) — unlike with ordinal variables, where we know a Bachelor’s is a bigger educational attainment than secondary/high school but there is no agreed unit to measure the “distance” precisely (as it’s neither measured in years, not in numbers of degrees).

Furthermore, my exam scores example lists 30, 65, 72… but I simply chose these numbers at random: I could have just as easily listed 25, 45, 70…, or 12, 54, 69…, etc. The point here is that one can have any number between 0 and 100 (in a conventional 100-point exam) as a potential score, or be a college student of any potential age (say, more than 5 years old),^[1] while the categories of an ordinal variable are fixed, or set, during operationalization (to a usually relatively small number), and cannot potentially be anything else (unless you operationalize the variable in a different way, which would result in a new variable). 

Finally, a happy corollary to the fact that interval and ratio variables’ categories are comprised of numbers is our ability to perform mathematical operations on them, beyond simple comparisons — something we can do neither with nominal, nor with ordinal variables. (Exactly what kind of mathematical operations we can do with interval and ratio variables you’ll see in Chapter 2.)

To summarize, interval and ratio variables have three defining features: 1) their categories (typically called values) are comprised of numbers, 2) the categories follow an order inherent in the fact that there is a measurable, unit-based scale, so that we can speak of a variable’s units of measurement, and 3) we can perform mathematical operations on the values (that the categories are).

Wait though… Why did I say that interval and ratio variables are different when I keep defining them together, and in the same way? Not to worry: the difference comes next, as I saved what students usually find the trickiest part for last.

With the risk of oversimplification (and, inevitably, exaggeration), interval scales are “made-up” while ratio scales are “real”. The difference is purely conceptual: you have to know whether the scale o which the variable is measured is “artificially designed”, as it were, or whether it exists as a some sort of “objective reality”. A rule-of-thumb advise on differentiating them that you may encounter is the “existence of a true zero”: ratio variables have a true zero while interval variables do not. (Clear as mud, eh? I did say it’s tricky.)

Examples usually help make this conundrum seem less of a conundrum.

In a similar vein, a score of 0 points on an exam doesn’t typically mean a complete absence of or no knowledge on a subject whatsoever — such a score usually simply means that the test-taker did not perform well on that particular test. Arguably, an easier test on the subject could be designed, and the test-taker would likely score more points.

Contrast this to our other variable from the original Do it! 1.2. exercise, age. Age of 0 years means exactly that – that we are talking about an infant who hasn’t yet reached their first birthday, and thus has completed 0 years of life (pardon the awkward phrasing). ^[3] Or consider a variable for, say, income: an income of $0 means the complete absence of income on dollars, i.e., no income. Both age and income are not “made up”: they exist regardless of how we measure them, and a zero on either indicates an absence of something (time in the former case, dollars in the latter). Physical attributes like height and weight work the same way.