Unlike a qualitative variable, a quantitative variable is one in which numerical level is a realization from some scale so that the distance between any two levels of the scale takes on meaning.
A continuous variable is one that can take on any value within a finite interval. For example, it is common to represent an insured’s age, weight, and income as continuous variables.
In contrast, a discrete variable is one that takes on only a finite number of values in any finite interval. For example, when examining a policyholder’s choice of deductibles, it may be that values of 0, 250, 500, and 1000 are the only possible outcomes. Like a ordinal variable, these represent distinct categories that are ordered. Unlike an ordinal variable, the numerical differences between variable levels take on (economic) meaning.
A special type of discrete variable is a count variable, one with values on the nonnegative integers (0, 1, 2, ldots.) For example, we will be particularly interested in the number of claims arising from a policy during a given period. This is known as the claim frequency.
Given that we will develop ways to analyze discrete variables, do we really need separate methods for dealing with continuous variables? After all, one can argue that few things in the physical world are truly continuous. For example, each currency has a smallest unit that is not subdivided further. (In the US, you cannot pay for anything smaller than one cent.) Nonetheless, models using continuous variables serve as excellent approximations to real-world discrete outcomes, in part due to their simplicity. It will be well worth our time and effort to develop models and analyze continuous and discrete variables differently.
Having said that, some variables inherently represent a combination of discrete and continuous components. An insurance claim is one such example. For an insurance claim, a discrete outcome at zero represents no insured loss, and a positive outcome represents the amount of the insured loss that we typically think of as continuous.
Another variable that features both discrete and continuous aspects is an interval variable, one that gives a range of possible outcomes. For example, instead of recording a driver’s age in year, it is common for insurers to group ages into three categories, (i) ages 16-24, representing young drives, (ii) ages 25-54, representing intermediate age drivers, and (iii) ages 55 and over, representing senior drivers. An interval variable is similar to an ordinal variable except, as in the driver’s age variable, we have some concept of distance between variable levels.
Circular data represent an interesting category typically not encountered by insurers. As an example of a circular variable, suppose that you monitor calls to your customer service center and would like to know when is the peak time of the day for calls to arrive. In this context, one can think about the time of the day as a variable with realizations on a circle, e.g., imagine an analog picture of a clock. For circular data, the distance between observations at 00:15 and 00:45 are just as close as observations 23:45 and 00:15 (here, we use the convention HH:MM to mean hours and minutes).