Continuous variable

A continuous variable is a quantitative variable whose possible values fill an interval. The working test is that between any two possible values there is always a third: name two heights and a height between them is possible, so the values never sit apart from each other in a list. That interval can be bounded. A proportion is continuous on 0 to 1 and a waiting time is continuous on an interval starting at 0, so continuous does not mean unlimited.

Five runners finish a 400 meter race in 58.2, 61.4, 59.9, 63.0 and 60.5 seconds. The times total 303.0, so the mean is $303.0/5 = 60.6$ seconds. Notice that the five recorded numbers step by tenths, because that is what the stopwatch reports, while the times themselves did not.

The wrong sentence is "the stopwatch only shows tenths, so time is discrete." That classifies the instrument instead of the variable. Refine the clock and you get more decimal places; refine your count of a household's cars and you get the same integer forever. Judge by the quantity, not by the precision of the record. Under that rule a height written to the nearest whole centimeter is still continuous, and a count of cars is discrete however you write it.

Because the values fill an interval, no single exact value carries positive probability once the variable is modeled with a curve. $P(T = 60.6)$ is 0, and only ranges such as $P(T < 60.6)$ carry any. That is why a continuous variable is displayed by binning values into a histogram or by a density curve whose total area is 1, and never by one bar per distinct value. Recorded values do repeat in real data sets, which is a fact about rounding rather than about the variable.

Topic 1.2 asks you to sort a quantitative variable into discrete or continuous. The continuous model you will meet most is the normal distribution, topic 2.11.