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I read in a book "An Introduction to Statistical Concepts [3 ed.] p.8):

A numerical variable is a quantitative variable. Numerical variables can further be classified as either discrete or continuous. A discrete variable is defined as a variable that can only take on certain values. For example, the number of children in a family can only take on certain values. Many values are not possible, such as negative values (e.g., the Joneses cannot have −2 children) or decimal values (e.g., the Smiths cannot have 2.2 children). In contrast, a continuous variable is defined as a variable that can take on any value within a certain range given a precise enough measurement instrument.

Question: Does this mean that a discrete variable cannot be a negative number? If a discrete variable cannot be a negative number then please explain why?

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    $\begingroup$ consider "$X_t$" is "number of goals scored in match $t$" and let $Y_t=X_t-X_{t-1}$. (i.e. the change in goals scored from the previous game). $Y_t$ is discrete but can clearly be negative. $\endgroup$ – Glen_b Mar 28 at 2:20
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    $\begingroup$ I guess in means in particular contexts certain values aren't possible. But in general a discrete variable could be negative or a decimal, irrational, etc etc. $\endgroup$ – innisfree Mar 28 at 8:27
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    $\begingroup$ @Glen_b: Or the more common "goal difference" defined as goals scored by one team versus goals scored by their opponent. The sign of this difference defines the winner of the game. $\endgroup$ – MSalters Mar 28 at 11:30
  • $\begingroup$ If I toss a coin 5 times the possible fractions of heads are 0/5, 1/5, ..., 4/5, 5/5 which are just as discrete as the corresponding counts 0 to 5. $\endgroup$ – Nick Cox Mar 29 at 0:13
  • $\begingroup$ The quoted definition is out-and-out wrong: it characterizes variables supported on a proper subset of the real numbers, not discrete variables. $\endgroup$ – whuber Mar 29 at 14:33
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Your intuition is correct -- a discrete variable can take on negative values.

The example is just an example: a person can't have $-2$ children, but the difference in scores between Home and Away sports teams can be $-2$ when the Home team is behind by two points.

Discrete variables with negative values exist all over the place. Two prominent examples:

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    $\begingroup$ (+1) To mention a somewhat simpler example: In certain sports (e.g. association football, ice hockey, Gaelic football, etc.) the goal difference between the home team and the visiting team can naturally modelled as a Skellam distribution. $\endgroup$ – usεr11852 Mar 28 at 23:29
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    $\begingroup$ @usεr11852 Yes, I should have turned to my knowledge of Gaelic football. ;-) Your point is well taken and I've revised the example. $\endgroup$ – Sycorax Mar 28 at 23:37
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The difference between continuous and discrete variables is not a mathematical essential one like the difference between natural and real numbers. It's just a matter of practicality: we use different tools to address each one because we are interested on answering different questions.

Basically, in discrete variables we are interested in the frequency of each value, but in continuous variables we are just interested in frequency of intervals. Then, we treat as continuous variables the variables when two or more cases getting the same value is just an anecdote - unlikely and/or uninteresting - and we model it as being able to get any real value in an interval. Otherwise, we model the variable as being a discrete variable with just a finite or numerable possible values.

For example: monetary quantities (prices, income, GDP and so) are usually modeled as continuous variables. However, they actually can only take a numerable set of values, because we just record monetary values up to some precision - usually 1 cent.

Some Euro area countries previous currency were valued less than 1 euro cent (e.g. Spanish peseta and Italian lira). In those countries cents had fallen in disuse long ago and all prices and wages were natural numbers, but when Euro was introduced they got a couple of decimal figures. Sometimes my students say that prices in pesetas were discrete variables but prices in euros are continuous ones, but that's plainly wrong because we are interested in the same questions and use the same statistical tools for both.

In summary and returning to the question: The difference between discrete an continuous variables are just a matter of convenience and you can treat a variable as discrete even if it takes negative values. You just need it to take few enough values to be interested in frequency of each one.

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